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Algebraic Expressions and Simplification
Building Blocks of Algebra
An algebraic expression is a mathematical phrase built from variables, constants, and operations (addition, subtraction, multiplication, division). Unlike a numerical expression that evaluates to a single number, an algebraic expression represents a general relationship that can take many values depending on the variables.
Variable: A symbol (usually $x$, $y$, $z$) that stands for an unknown or changeable value.
Constant: A fixed number such as $5$, $-3$, or $\frac{1}{2}$ that never changes within the expression.
Coefficient: The numerical factor multiplied by a variable — in $7x^2$, the coefficient of $x^2$ is $7$.
Term: A single number, variable, or product of numbers and variables separated by $+$ or $-$ signs. The expression $3x^2 - 5x + 8$ has three terms.
Understanding like terms and unlike terms is essential for combining and simplifying expressions. Like terms share the same variable part — they differ only in their coefficient.
$$3x^2y \;\text{and} \;-7x^2y \;\text{are like terms;} \quad 3x^2y \;\text{and} \;3xy^2 \;\text{are not}$$
Same Variable Part: Like terms must have exactly the same variables raised to exactly the same powers. $2ab^2$ and $-5ab^2$ are like; $2ab^2$ and $2a^2b$ are not.
Coefficients Can Differ: Only the number in front changes — $4x$ and $-9x$ are like terms and can be combined to $-5x$.
Constants Are Always Like: All standalone numbers are like terms with each other since they share the same (empty) variable part.
Exponent Check: $x^2$ and $x^3$ are unlike terms because the powers differ, even though the variable is the same.
Expressions are classified by the number of terms they contain and by their degree — the highest sum of exponents on variables in any single term.
Monomial: One term — $7x^3$, $-2ab$, $15$.
Binomial: Two terms — $3x + 5$, $x^2 - 4y$, $a^3 + b^3$.
Trinomial: Three terms — $x^2 + 5x + 6$, $2a^2 - 3ab + b^2$.
Polynomial: Any expression with one or more terms where all exponents are non-negative integers.
Degree of a Term: Sum of exponents on all variables — $3x^2y^3$ has degree $5$.
Degree of a Polynomial: The highest degree among its terms — $4x^3 + 2x^2 - x + 1$ is degree $3$.
Standard Polynomial Terminology
•
Degree 0: constant (e.g., 7)
•
Degree 1: linear (e.g., 3x + 2)
•
Degree 2: quadratic (e.g., x² − 5x + 6)
•
Degree 3: cubic (e.g., 2x³ − x + 1)
•
Degree 4: quartic (e.g., x⁴ − 3x² + 1)
Operations on Algebraic Expressions
Adding and subtracting expressions comes down to removing brackets and combining like terms. The key is handling the sign that precedes each bracket correctly.
Addition: Remove brackets and combine like terms. $(3x + 2) + (5x - 7) = 8x - 5$.
Subtraction: Distribute the negative sign to EVERY term inside the brackets, then combine. $(4x^2 + 3x) - (2x^2 - x + 5) = 4x^2 + 3x - 2x^2 + x - 5 = 2x^2 + 4x - 5$.
Vertical Method: For longer expressions, stack terms in columns by like terms and add or subtract vertically — this reduces errors.
Parentheses Within Parentheses: Work from the innermost brackets outward. Simplify $(2x + (3x - 1) - (x + 4))$ by starting inside.
Multiplying and dividing expressions relies on the distributive property and the laws of exponents. When multiplying two polynomials, every term in the first must multiply every term in the second.
$$(a + b)(c + d) = ac + ad + bc + bd$$
Monomial × Polynomial: Multiply the monomial through each term. $3x(2x^2 - 4x + 1) = 6x^3 - 12x^2 + 3x$.
FOIL for Binomials: First, Outer, Inner, Last — $(x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15$.
General Polynomial Multiplication: Use a table or the column method — each term of the first polynomial multiplies each term of the second.
Division of a Polynomial by a Monomial: Divide each term separately. $\frac{6x^3 + 9x^2}{3x} = 2x^2 + 3x$.
Expansion of Brackets
The distributive property is the foundation of bracket expansion — it lets you convert a product of a factor and a sum into a sum of individual products.
$$a(b + c + d) = ab + ac + ad$$
Multiply the outside factor with every term inside the brackets.
$a$=Factor outside the brackets(any expression)
$b, c, d$=Terms inside the brackets(any expressions)
$a = -1$
→Every term inside flips sign: $-(b + c - d) = -b - c + d$.
Single Bracket: $-2x(3x^2 - 4x + 7) = -6x^3 + 8x^2 - 14x$. The negative sign flips every term.
Nested Brackets: Expand from the inside out. $2(x + 3(x - 1)) = 2(x + 3x - 3) = 2(4x - 3) = 8x - 6$.
Bracket × Bracket: Each term of the first bracket multiplies each term of the second — this is generalized FOIL.
Squared brackets like $(a + b)^2$ are NOT simply $a^2 + b^2$. You must expand them using the binomial square formulas, which are special cases of the algebraic identity family.
$$(a + b)^2 = a^2 + 2ab + b^2$$
The square of a sum equals the sum of squares plus twice the product.
$a^2$=Square of the first term(any expression)
$2ab$=Twice the product of both terms(any expression)
$b^2$=Square of the second term(any expression)
$b = 0$
→Reduces to $a^2$.
$a = b$
→$(2a)^2 = 4a^2$, confirming the middle term doubles the cross product.
The Middle Term: The $2ab$ term is what students forget — it accounts for the Outer and Inner products in FOIL.
Subtraction Variant: $(a - b)^2 = a^2 - 2ab + b^2$. The middle term is negative, but $b^2$ stays positive.
Quick Mental Math: $(x + 7)^2 = x^2 + 14x + 49$ — the middle coefficient is twice the constant.
Core Algebraic Identities
The difference of squares is one of the most frequently used algebraic identity in simplification and factorization. It converts a product of a sum and difference into the difference of two squares.
$$a^2 - b^2 = (a + b)(a - b)$$
The difference of two perfect squares factors into a conjugate pair — the same terms with opposite signs between them.
$a^2$=First perfect square(any expression squared)
$b^2$=Second perfect square(any expression squared)
$(a+b)$=Sum of the square roots(any expressions)
$(a-b)$=Difference of the square roots(any expressions)
$a = b$
→$a^2 - a^2 = 0 = (a+a)(a-a) = 2a \cdot 0 = 0$.
$a = 1, b = x$
→$1 - x^2 = (1+x)(1-x)$ — used to rationalize denominators.
Rationalizing Denominators: Multiply numerator and denominator by the conjugate to eliminate radicals: $\frac{1}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{\sqrt{3} + 1}{3 - 1} = \frac{\sqrt{3} + 1}{2}$.
Chain Application: $a^4 - b^4 = (a^2)^2 - (b^2)^2 = (a^2 + b^2)(a^2 - b^2) = (a^2 + b^2)(a+b)(a-b)$.
Spotting the Pattern: Any expression of the form (something)$^2$ − (something)$^2$ can use this identity.
The sum and difference of cubes factor into a linear factor times a quadratic factor. The quadratic factor always has the 'wrong' sign in the middle term.
$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
Sum of cubes factors into a linear binomial and a quadratic trinomial where the middle term has the opposite sign.
$(a+b)$=Linear factor — same sign as the original(any expressions)
$a^2 - ab + b^2$=Quadratic factor — middle term has the FLIPPED sign(any expressions)
$b = 0$
→$a^3 = a(a^2)$ — trivial case.
Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ — the linear factor takes the original sign, the quadratic flips it.
Sign Rule (SOAP): Same sign, Opposite sign, Always Positive — the linear factor has the Same sign as the original, the middle term of the quadratic has the Opposite sign, and the last term is Always Positive.
Recognizing Cubes: $8x^3 = (2x)^3$, $27y^3 = (3y)^3$, $64 = 4^3$ — identify the cube roots first.
Common Cubes to Recognize
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1 = 1³, 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³
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216 = 6³, 343 = 7³, 512 = 8³, 729 = 9³, 1000 = 10³
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$x^3$, $8x^3 = (2x)^3$, $27x^3 = (3x)^3$, $125x^3 = (5x)^3$
A perfect square trinomial is the result of squaring a binomial. Recognizing one lets you reverse the process and factor it instantly.
$$x^2 \pm 2ax + a^2 = (x \pm a)^2$$
A trinomial is a perfect square when the first and last terms are perfect squares and the middle term is twice the product of their roots.
$x^2$=Square of the variable(any expression squared)
$a^2$=Square of the constant(any expression squared)
$\pm 2ax$=Twice the product of roots (with the correct sign)(any expression)
Recognition Test: Check if $\text{middle term} = 2\sqrt{\text{first term} \times \text{last term}}$. For $x^2 + 10x + 25$: $2\sqrt{x^2 \cdot 25} = 2 \cdot 5x = 10x$ ✓
Leading Coefficient Not 1: $4x^2 + 12x + 9 = (2x)^2 + 2(2x)(3) + 3^2 = (2x + 3)^2$.
Completing the Square: Given $x^2 + 6x$, add $9$ to form $x^2 + 6x + 9 = (x+3)^2$.
Factorization Techniques
The first step in any factorization is extracting the greatest common factor (GCF) — the largest expression that divides every term of the polynomial.
$$ab + ac + ad = a(b + c + d)$$
Factor out the GCF to simplify the expression and reveal the remaining structure.
$a$=Greatest common factor of all terms(any expression)
$b + c + d$=Remaining factor after dividing each term by the GCF(any expressions)
Numerical GCF: For $6x^2 + 9x$, the GCF of $6$ and $9$ is $3$, so factor out $3x$: $3x(2x + 3)$.
Variable GCF: For $x^3y - x^2y^2 + xy^3$, the GCF is $xy$: $xy(x^2 - xy + y^2)$.
Always Try First: Before attempting any other technique, remove the GCF. It simplifies every subsequent step.
Negative GCF: Sometimes factoring out $-1$ helps, especially when the leading coefficient is negative: $-x^2 + 4x - 3 = -(x^2 - 4x + 3)$.
When a polynomial has four or more terms and no common factor across all of them, grouping rearranges terms into pairs that each have a common factor, revealing a shared binomial factor.
Standard Grouping: $x^3 + 3x^2 - 4x - 12 = x^2(x + 3) - 4(x + 3) = (x^2 - 4)(x + 3) = (x+2)(x-2)(x+3)$.
Rearranging Terms: Sometimes the natural order doesn't group well. Try rearranging: $2x^2 + 3x - 4x - 6 = x(2x + 3) - 2(2x + 3) = (x - 2)(2x + 3)$.
Splitting the Middle Term: For $ax^2 + bx + c$, find two numbers $p$ and $q$ such that $p + q = b$ and $pq = ac$. Then split the middle: $ax^2 + px + qx + c$ and group.
When Grouping Fails: Not every polynomial can be factored by grouping — if no rearrangement produces a common binomial factor, try other methods.
Factoring a quadratic trinomial $ax^2 + bx + c$ means finding two binomials whose product equals the original expression. This is the reverse of FOIL.
$$x^2 + (p+q)x + pq = (x + p)(x + q)$$
Find two numbers that add to $b$ (the middle coefficient) and multiply to $c$ (the constant term).
$p + q$=Sum of the two numbers = coefficient of $x$(any numbers)
$pq$=Product of the two numbers = constant term(any numbers)
$c > 0, b > 0$
→Both $p$ and $q$ are positive.
$c > 0, b < 0$
→Both $p$ and $q$ are negative.
$c < 0$
→$p$ and $q$ have opposite signs.
Leading Coefficient 1: $x^2 + 7x + 12$ — find two numbers adding to $7$ and multiplying to $12$: $3$ and $4$. So $(x+3)(x+4)$.
Leading Coefficient Not 1 ($ac$ method): For $6x^2 + 11x + 3$, multiply $ac = 18$. Find two numbers adding to $11$ with product $18$: $9$ and $2$. Split: $6x^2 + 9x + 2x + 3 = 3x(2x+3) + 1(2x+3) = (3x+1)(2x+3)$.
Discriminant Check: Before factoring, compute $b^2 - 4ac$. If it's not a perfect square, the quadratic doesn't factor over the integers.
Simplification Strategies
Simplifying algebraic fractions involves factoring numerator and denominator, cancelling common factors, and performing operations just like arithmetic fractions.
$$\frac{a^2 - b^2}{a^2 - 2ab + b^2} = \frac{(a+b)(a-b)}{(a-b)^2} = \frac{a+b}{a-b}$$
Factor Then Cancel: Always factor numerator and denominator before cancelling — never cancel individual terms within a sum. $\frac{x + 3}{x}$ does NOT simplify to $3$.
Multiplication: Multiply numerators together and denominators together, then simplify. $\frac{x+1}{x} \cdot \frac{x}{x-1} = \frac{(x+1)x}{x(x-1)} = \frac{x+1}{x-1}$.
Addition with Unlike Denominators: Find the least common denominator (LCD) by factoring each denominator, then convert and combine. $\frac{1}{x+1} + \frac{1}{x-1} = \frac{(x-1)+(x+1)}{(x+1)(x-1)} = \frac{2x}{x^2-1}$.
Complex Fractions: Simplify the numerator and denominator separately, then multiply by the reciprocal of the denominator.
Many problems involve evaluating or simplifying expressions by substitution — replacing variables with given values or using substitution to reveal hidden structure.
Direct Substitution: Plug in values and compute: If $x = 2$ and $y = -3$, then $3x^2 - 2xy + y^2 = 3(4) - 2(2)(-3) + 9 = 12 + 12 + 9 = 33$.
Strategic Substitution: In expressions like $\frac{1}{x + \frac{1}{y + \frac{1}{z}}}$, work from the innermost fraction outward.
Expression Evaluation Without Full Expansion: To evaluate $(x + 3)^2 - (x - 3)^2$ at $x = 5$, expand using identities: $[(x+3)-(x-3)][(x+3)+(x-3)] = (6)(2x) = 12x = 60$.
Symmetric Expressions: For $x + \frac{1}{x} = 3$, find $x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 = 9 - 2 = 7$.
Application Strategy
Worked Examples
Problem Statement
A store's revenue in thousands of dollars is modeled by $3x^2 - 5x + 8$ for product A and $2x^2 + 3x - 4$ for product B. Find the total revenue expression.
like terms
Strategic Toolkit
$$R = (3x^2 - 5x + 8) + (2x^2 + 3x - 4)$$
Step 1
Remove brackets
Since we're adding, remove the brackets and write all terms together.
$3x^2 - 5x + 8 + 2x^2 + 3x - 4$
Step 2
Group like terms
Group terms that share the same variable part.
$(3x^2 + 2x^2) + (-5x + 3x) + (8 - 4)$
Step 3
Combine coefficients
Combine the coefficients within each group.
$$5x^2 - 2x + 4$$
Problem Statement
A rectangular garden has length $(3x + 5)$ meters and width $(2x - 3)$ meters. Find the area as an expanded expression.
distributive property
Strategic Toolkit
$$A = (3x + 5)(2x - 3)$$
Step 1
Apply FOIL
Apply the FOIL method: multiply First, Outer, Inner, and Last pairs.
$3x \cdot 2x = 6x^2, \quad 3x \cdot (-3) = -9x, \quad 5 \cdot 2x = 10x, \quad 5 \cdot (-3) = -15$
Step 2
Write all products
Write out all four products from the expansion.
$$6x^2 - 9x + 10x - 15$$
Step 3
Combine like terms
Combine the like terms and state the final area expression.
$$6x^2 + x - 15$$
Problem Statement
Simplify $\frac{x^2 - 49}{x^2 + 14x + 49}$.
difference of squaresfactorization
Strategic Toolkit
$$\frac{x^2 - 49}{x^2 + 14x + 49} = \frac{(x+7)(x-7)}{(x+7)^2}$$
Step 1
Factor the numerator
Recognize the numerator as a difference of squares and factor it.
$$x^2 - 49 = (x+7)(x-7)$$
Step 2
Factor the denominator
Recognize the denominator as a perfect square trinomial and factor it.
$$x^2 + 14x + 49 = (x+7)^2$$
Step 3
Cancel the common factor
Cancel the common binomial factor from numerator and denominator.
$$\frac{x - 7}{x + 7}$$
Problem Statement
Factor $6x^2 + 7x - 20$ completely.
factorization
Strategic Toolkit
$$6x^2 + 7x - 20 = (3x - 4)(2x + 5)$$
Step 1
Identify a, b, c
Identify the coefficients and compute the product ac.
$a = 6, \quad b = 7, \quad c = -20, \quad ac = 6 \times (-20) = -120$
Step 2
Find two numbers
Find two numbers that multiply to ac and add to b.
$p + q = 7, \quad pq = -120 \implies p = 15, \quad q = -8$
Step 3
Split the middle term
Rewrite the middle term using the two numbers found.
$$6x^2 + 15x - 8x - 20$$
Step 4
Factor by grouping
Factor out the common binomial from each group.
$$(3x - 4)(2x + 5)$$
Problem Statement
Factor $x^3 - 2x^2 - 9x + 18$ completely.
groupingfactorization
Strategic Toolkit
$$x^3 - 2x^2 - 9x + 18 = (x - 2)(x + 3)(x - 3)$$
Step 1
Group in pairs
Split the four terms into two pairs.
$(x^3 - 2x^2) + (-9x + 18)$
Step 2
Factor each pair
Factor out the greatest common factor from each pair.
$x^2(x - 2) - 9(x - 2)$
Step 3
Extract common binomial
Extract the common binomial factor.
$(x^2 - 9)(x - 2)$
Step 4
Factor the difference of squares
Factor the remaining difference of squares.
$$(x+3)(x-3)(x-2)$$
Problem Statement
If $a + b = 12$ and $ab = 35$, find the value of $a^2 + b^2$.
algebraic identity
Strategic Toolkit
$$a^2 + b^2 = (a+b)^2 - 2ab$$
Step 1
Start with the identity
Rearrange the binomial square identity to express the sum of squares.
$a^2 + b^2 = (a+b)^2 - 2ab$
Step 2
Substitute known values
Substitute the known values of a+b and ab.
$$a^2 + b^2 = 144 - 70 = 74$$
Problem Statement
Simplify $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$.
algebraic fraction
Strategic Toolkit
$$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}} = \frac{y + x}{y - x}$$
Step 1
Find common denominators
Rewrite each fraction with the common denominator xy, then combine.
$\frac{y}{xy} + \frac{x}{xy} = \frac{x+y}{xy}, \quad \frac{y}{xy} - \frac{x}{xy} = \frac{y-x}{xy}$
Step 2
Divide by the denominator
Divide the numerator fraction by the denominator fraction by multiplying by the reciprocal.
$$\frac{x+y}{y-x}$$
Step 3
Simplify
The simplified form can also be written with a negative sign to flip the denominator.
$-\frac{x+y}{x-y}$
Problem Statement
Expand and simplify $(2x + 1)^3$.
algebraic identitydistributive property
Strategic Toolkit
$$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$
Step 1
Identify a and b
Match the expression to the identity pattern by identifying a and b.
$a = 2x, \quad b = 1$
Step 2
Apply the cubic identity
Apply the cubic expansion identity.
$(2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + (1)^3$
Step 3
Compute each term
Compute each term of the expansion.
$$8x^3 + 12x^2 + 6x + 1$$
Memorize formulas
Binomial Square (Sum)
Expanding the square of a sum
Formula
$(a+b)^2 = a^2 + 2ab + b^2$
Binomial Square (Difference)
Expanding the square of a difference
Formula
$(a-b)^2 = a^2 - 2ab + b^2$
Difference of Squares
Factoring or expanding a² − b²
Formula
$a^2 - b^2 = (a+b)(a-b)$
Cube of a Sum
Expanding (a+b)³
Formula
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
Cube of a Difference
Expanding (a-b)³
Formula
$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
Sum of Cubes
Factoring a³ + b³
Formula
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of Cubes
Factoring a³ − b³
Formula
$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
Distributive Property
Expanding brackets — multiply each term inside
Formula
$a(b+c) = ab + ac$
Review
1
Which of the following pairs are like terms?
Ans
$3x^2y$ and $-5x^2y$
Review
Tap to reveal
Check each pair for matching variable parts
Two terms are like terms only when every variable appears with the same exponent in both terms. The coefficients do not need to match.
$$3x^2y \text{ and } -5x^2y \text{ — same variables } x^2, y \text{ ✓}$$
2
What is the degree of the polynomial $5x^3 - 2x^2y + 7xy^2 - y^3 + 4$?
Ans
$3$
Review
Tap to reveal
Find the degree of each term
Sum the exponents in each term: constants have degree $0$, and each variable's exponent contributes to the total.
$$5x^3\!:\ 3,\quad -2x^2y\!:\ 3,\quad 7xy^2\!:\ 3,\quad -y^3\!:\ 3,\quad 4\!:\ 0$$
Identify the maximum
The degree of the polynomial equals the highest degree among its terms.
$$\text{Degree} = 3$$
3
Expand and simplify: $3(2x - 4) - 2(x + 5)$
Ans
$4x - 22$
Review
Tap to reveal
Apply the distributive property to both groups
Multiply each term inside each set of brackets by the factor outside.
$$3(2x - 4) = 6x - 12, \quad -2(x + 5) = -2x - 10$$
Combine like terms
Group the $x$-terms and the constant terms separately.
$$6x - 2x - 12 - 10 = 4x - 22$$
4
Simplify: $4x^2 - 3x + 2 - 5x^2 + 7x - 8 + x^2$
Ans
$4x - 6$
Review
Tap to reveal
Group and combine the $x^2$ terms
Collect all terms with $x^2$ and sum their coefficients.
$$4x^2 - 5x^2 + x^2 = (4 - 5 + 1)x^2 = 0$$
Group and combine remaining terms
Collect the $x$-terms and constant terms separately.
$$-3x + 7x = 4x, \quad 2 - 8 = -6$$
5
If $x + y = 8$ and $xy = 15$, what is $x^2 + y^2$?
Ans
$34$
Review
Tap to reveal
Apply the binomial square identity
Expand $(x+y)^2$ using the identity to express $x^2 + y^2$ in terms of known quantities.
$$(x + y)^2 = x^2 + 2xy + y^2$$
Substitute the given values and solve
Replace $(x+y)^2$ with $64$ and $2xy$ with $30$, then isolate $x^2 + y^2$.
$$64 = x^2 + y^2 + 30 \implies x^2 + y^2 = 34$$
6
Factorize $144x^2 - 25$.
Ans
$(12x + 5)(12x - 5)$
Review
Tap to reveal
Recognize the difference of squares pattern
Express each term as a perfect square: $144x^2 = (12x)^2$ and $25 = 5^2$.
$$144x^2 - 25 = (12x)^2 - 5^2$$
Apply the identity $a^2 - b^2 = (a+b)(a-b)$
Factor using the difference of squares formula with $a = 12x$ and $b = 5$.
$$(12x + 5)(12x - 5)$$
7
Factorize $x^2 - 7x + 12$.
Ans
$(x - 3)(x - 4)$
Review
Tap to reveal
Find the factor pair
List pairs of integers that multiply to $12$ and check which pair sums to $-7$.
$$-3 \times -4 = 12, \quad -3 + (-4) = -7$$
Write the factors
Use the two numbers as constants in the binomial factors.
$$x^2 - 7x + 12 = (x - 3)(x - 4)$$
8
A square garden has side length $(x + 3)$ meters. A smaller square pond inside it has side length $(x - 2)$ meters. What is the area of the garden excluding the pond?
Ans
$10x + 5$
Review
Tap to reveal
Set up the area expression
The shaded area is the difference between the large square's area and the pond's area.
$$(x+3)^2 - (x-2)^2$$
Expand using the binomial square identity
Apply $(a \pm b)^2 = a^2 \pm 2ab + b^2$ to each square, then subtract.
$$(x^2 + 6x + 9) - (x^2 - 4x + 4) = 10x + 5$$
9
A rectangular field has length $(l + w)$ meters and width $(l - w)$ meters. If $l = 13$ and $w = 5$, what is the area of the field?
Ans
$144 \text{ m}^2$
Review
Tap to reveal
Apply the difference of squares identity
The area $(l+w)(l-w)$ matches the identity $a^2 - b^2 = (a+b)(a-b)$ with $a = l$ and $b = w$.
$$(l+w)(l-w) = l^2 - w^2$$
Substitute the values
Replace $l$ with $13$ and $w$ with $5$.
$$13^2 - 5^2 = 169 - 25 = 144 \text{ m}^2$$
10
A shopkeeper buys $(3x + 2)$ crates of supplies at Rs. $(2x - 1)$ per crate. Which expression gives the total cost?
Ans
$6x^2 + x - 2$
Review
Tap to reveal
Set up the multiplication
Total cost equals number of crates times price per crate.
$$(3x + 2)(2x - 1)$$
Expand using FOIL
Multiply each term in the first bracket by each term in the second.
$$6x^2 - 3x + 4x - 2 = 6x^2 + x - 2$$
11
If $(a - b)^2 = 49$ and $ab = 10$, find the value of $a^2 + b^2$.
Ans
$69$
Review
Tap to reveal
Write out the binomial square identity
Expand $(a-b)^2$ and substitute the known values.
$$(a-b)^2 = a^2 - 2ab + b^2$$
Substitute known values and isolate $a^2 + b^2$
Replace $(a-b)^2$ with $49$ and $2ab$ with $20$, then rearrange.
$$49 = a^2 + b^2 - 20 \implies a^2 + b^2 = 69$$
12
If $a + b = 5$ and $a^2 + b^2 = 13$, find $a^3 + b^3$.
Ans
$35$
Review
Tap to reveal
Find $ab$ using the binomial square identity
From $(a+b)^2 = a^2 + 2ab + b^2$, solve for $ab$.
$$25 = 13 + 2ab \implies ab = 6$$
Apply the sum of cubes identity
Use $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ and substitute all known values.
$$5 \times (13 - 6) = 5 \times 7 = 35$$
13
A cube has side length $x$ cm. When each side is increased by $2$ cm, the new volume exceeds the old volume by $98$ cm³. Find $x$.
Ans
$3$
Review
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Set up the equation from the volume difference
The increase in volume equals the new cube volume minus the original cube volume.
$$(x+2)^3 - x^3 = 98$$
Expand and simplify
Use the cube expansion identity and cancel the $x^3$ terms.
$$6x^2 + 12x + 8 = 98 \implies x^2 + 2x - 15 = 0$$
Solve the quadratic and select the valid root
Factor the quadratic and choose the positive solution since side length must be positive.
$$(x+5)(x-3) = 0 \implies x = 3$$
14
Factorize $x^3 - 3x^2 + 5x - 15$ by grouping.
Ans
$(x - 3)(x^2 + 5)$
Review
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Group terms in pairs
Split the four-term expression into two groups that share a common binomial factor.
$$(x^3 - 3x^2) + (5x - 15)$$
Factor each group and extract the common binomial
Factor out the GCF from each group, then factor out the shared binomial $(x-3)$.
$$x^2(x - 3) + 5(x - 3) = (x^2 + 5)(x - 3)$$
15
A rectangular banner has area expressed as $\frac{x^2 - 9}{x + 3}$ square meters. Which simplified expression represents the width if the length is $x + 3$ meters (for $x \neq -3$)?
Ans
$x - 3$
Review
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Factor the numerator as a difference of squares
Recognize $x^2 - 9 = (x+3)(x-3)$.
$$\frac{(x + 3)(x - 3)}{x + 3}$$
Cancel the common factor
Divide numerator and denominator by $(x+3)$, valid when $x \neq -3$.
$$x - 3$$
16
Factorize $8x^3 - 27y^3$ completely.
Ans
$(2x - 3y)(4x^2 + 6xy + 9y^2)$
Review
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Identify the cube terms
Express each part as a perfect cube to match the difference of cubes formula.
$$8x^3 = (2x)^3, \quad 27y^3 = (3y)^3$$
Apply the difference of cubes identity
Use $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ with $a = 2x$ and $b = 3y$.
$$(2x - 3y)(4x^2 + 6xy + 9y^2)$$
17
What is $103^2 - 97^2$?
Ans
$1200$
Review
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Apply the difference of squares identity
Rather than computing two large squares, use $a^2 - b^2 = (a+b)(a-b)$ for quick mental arithmetic.
$$103^2 - 97^2 = (103 + 97)(103 - 97)$$
Compute the product
Evaluate the sum and difference, then multiply.
$$200 \times 6 = 1200$$
18
A factory produces $p$ units per day for $5$ days and $(p - 20)$ units per day for the next $3$ days. If total output is $860$ units, what is $p$?
Ans
$115$
Review
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Write the total production expression
Multiply daily output by the number of days for each period and add.
$$5p + 3(p - 20) = 860$$
Solve for $p$
Expand, combine like terms, and isolate $p$.
$$8p - 60 = 860 \implies 8p = 920 \implies p = 115$$
19
Factorize $x^4 - 16$ completely over the integers.
Ans
$(x^2 + 4)(x + 2)(x - 2)$
Review
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Apply difference of squares on the $x^4$ term
Recognize $x^4 - 16 = (x^2)^2 - 4^2$ and factor as a difference of squares.
$$(x^2 + 4)(x^2 - 4)$$
Factor $x^2 - 4$ further
The factor $x^2 - 4$ is itself a difference of squares and can be broken down.
$$(x^2 + 4)(x + 2)(x - 2)$$
20
A rectangular plot has length $l$ meters and width $w$ meters. The perimeter is $60$ m and the area is $200$ m². Find $l^2 + w^2$.
Ans
$500$
Review
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Use the perimeter to find $l + w$
The perimeter formula $2(l+w) = 60$ gives the sum of length and width.
$$l + w = 30$$
Apply the binomial square identity
Use $(l+w)^2 = l^2 + 2lw + w^2$ with the known values of $l+w$ and $lw$.
$$900 = l^2 + w^2 + 2(200) \implies l^2 + w^2 = 500$$
Question 1 of 20
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1
Which of the following pairs are like terms?