ASVAB Study Guide
Pre-Algebra · Lesson 2 of 3

Multiplying Polynomials

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Monomial by Polynomial

Multiplying a single term by multiple terms

$$a \times (b + c + \ldots) = ab + ac + \ldots$$

Use the Distributive Property. Multiply the single term by every term inside the parentheses.

Example

Multiply $$3x^2 \times (2x - 5)$$

Show solution

Step 1: Distribute $$3x^2$$ to each term inside:

  • $$3x^2 \times 2x = (3 \times 2) \times (x^2 \times x) = 6x^3$$
  • $$3x^2 \times (-5) = -15x^2$$

Step 2: Combine terms.

Result: $$6x^3 - 15x^2$$

Practice problems

  1. Problem 1

    Multiply $$4x \times (x + 3)$$

    Show solution

    Answer: $$4x^2 + 12x$$

    Step 1: Distribute $$4x$$ to x: $$4x \times x = 4x^2$$.

    Step 2: Distribute $$4x$$ to 3: $$4x \times 3 = 12x$$.

    => Combined result $$= 4x^2 + 12x$$.

    => Correct Answer: A ($$4x^2 + 12x$$)

  2. Problem 2

    Multiply $$-2x^2 \times (3x^2 - 4x + 1)$$

    Show solution

    Answer: $$-6x^4 + 8x^3 - 2x^2$$

    Step 1: $$-2x^2 \times 3x^2 = -6x^4$$.

    Step 2: $$-2x^2 \times (-4x) = +8x^3$$.

    Step 3: $$-2x^2 \times 1 = -2x^2$$.

    => Result $$= -6x^4 + 8x^3 - 2x^2$$.

    => Correct Answer: A ($$-6x^4 + 8x^3 - 2x^2$$)

  3. Problem 3

    Multiply $$5x y^2 \times (2x^2 y - 3xy + 4)$$.

    Show solution

    Answer: $$10x^3 y^3 - 15x^2 y^3 + 20x y^2$$

    Step 1: $$5x y^2 \times 2x^2 y = 10 x^{(1+2)} y^{(2+1)} = 10x^3 y^3$$.

    Step 2: $$5x y^2 \times (-3xy) = -15 x^{(1+1)} y^{(2+1)} = -15x^2 y^3$$.

    Step 3: $$5x y^2 \times 4 = 20x y^2$$.

    => Result $$= 10x^3 y^3 - 15x^2 y^3 + 20x y^2$$.

    => Correct Answer: B ($$10x^3 y^3 - 15x^2 y^3 + 20x y^2$$)

2. Binomial by Binomial

Multiplying two expressions with two terms each

$$(a + b)(c + d) = ac + ad + bc + bd$$

Use the FOIL method: First, Outer, Inner, Last. Combine like terms at the end.

Example

Multiply $$(x + 3)(x - 4)$$

Show solution

Step 1: First: $$x \times x = x^2$$

Step 2: Outer: $$x \times (-4) = -4x$$

Step 3: Inner: $$3 \times x = 3x$$

Step 4: Last: $$3 \times (-4) = -12$$

Step 5: Combine middle terms: $$-4x + 3x = -x$$

Result: $$x^2 - x - 12$$

Practice problems

  1. Problem 4

    Multiply $$(x + 2)(x + 5)$$

    Show solution

    Answer: $$x^2 + 7x + 10$$

    Step 1: First = $$x^2$$, Outer $$= 5x$$, Inner $$= 2x$$, Last = 10.

    Step 2: Combine Outer and Inner: $$5x + 2x = 7x$$.

    => Result $$= x^2 + 7x + 10$$.

    => Correct Answer: A ($$x^2 + 7x + 10$$)

  2. Problem 5

    Multiply $$(2x - 3)(3x + 4)$$

    Show solution

    Answer: $$6x^2 - x - 12$$

    Step 1: First = $$2x \times 3x = 6x^2$$.

    Step 2: Outer $$= 2x \times 4 = 8x$$.

    Step 3: Inner $$= -3 \times 3x = -9x$$.

    Step 4: Last $$= -3 \times 4 = -12$$.

    Step 5: Combine middle terms: $$8x - 9x = -x$$.

    => Result $$= 6x^2 - x - 12$$.

    => Correct Answer: A ($$6x^2 - x - 12$$)

  3. Problem 6

    Multiply $$(4x - 5y)(2x - 3y)$$

    Show solution

    Answer: $$8x^2 - 22xy + 15y^2$$

    Step 1: First = $$4x \times 2x = 8x^2$$.

    Step 2: Outer $$= 4x \times (-3y) = -12xy$$.

    Step 3: Inner $$= -5y \times 2x = -10xy$$.

    Step 4: Last $$= -5y \times (-3y) = +15y^2$$.

    Step 5: Combine middle terms: $$-12xy - 10xy = -22xy$$.

    => Result $$= 8x^2 - 22xy + 15y^2$$.

    => Correct Answer: A ($$8x^2 - 22xy + 15y^2$$).

3. Special Products

Squaring a binomial (Perfect Square)

$$(a \pm b)^2 = a^2 \pm 2ab + b^2$$

Square the first term, add twice the product of both terms, then square the last term.

Example

Expand $$(2x - 3)^2$$

Show solution

Step 1: Square first term: $$(2x)^2 = 4x^2$$.

Step 2: Twice product: $$2 \times (2x) \times (-3) = -12x$$.

Step 3: Square last term: $$(-3)^2 = 9$$.

Result: $$4x^2 - 12x + 9$$

Practice problems

  1. Problem 7

    Expand $$(x + 4)^2$$

    Show solution

    Answer: $$x^2 + 8x + 16$$

    Step 1: First term squared = $$x^2$$.

    Step 2: Twice product $$= 2 \times x \times 4 = 8x$$.

    Step 3: Last term squared $$= 4^2 = 16$$.

    => Result $$= x^2 + 8x + 16$$.

    => Correct Answer: A ($$x^2 + 8x + 16$$)

  2. Problem 8

    Expand $$(3x - 5)^2$$

    Show solution

    Answer: $$9x^2 - 30x + 25$$

    Step 1: First term squared = $$(3x)^2 = 9x^2$$.

    Step 2: Twice product $$= 2 \times (3x) \times (-5) = -30x$$.

    Step 3: Last term squared $$= (-5)^2 = 25$$.

    => Result $$= 9x^2 - 30x + 25$$.

    => Correct Answer: A ($$9x^2 - 30x + 25$$)

  3. Problem 9

    Expand $$(4x^2 + 3y)^2$$

    Show solution

    Answer: $$16x^4 + 24x^2 y + 9y^2$$

    Step 1: First term squared = $$(4x^2)^2 = 16x^4$$.

    Step 2: Twice product $$= 2 \times (4x^2) \times (3y) = 24x^2 y$$.

    Step 3: Last term squared $$= (3y)^2 = 9y^2$$.

    => Result $$= 16x^4 + 24x^2 y + 9y^2$$.

    => Correct Answer: A ($$16x^4 + 24x^2 y + 9y^2$$)

4. Difference of Squares

Multiplying sum and difference of same terms

$$(a + b)(a - b) = a^2 - b^2$$

The middle terms (Outer and Inner) cancel each other out, leaving only the squares.

Example

Multiply $$(3x + 5)(3x - 5)$$

Show solution

Step 1: Square first term: $$(3x)^2 = 9x^2$$.

Step 2: Square second term: $$5^2 = 25$$.

Step 3: Subtract: $$9x^2 - 25$$.

Result: $$9x^2 - 25$$

Practice problems

  1. Problem 10

    Multiply $$(x + 6)(x - 6)$$

    Show solution

    Answer: $$x^2 - 36$$

    Step 1: Square first term = $$x^2$$.

    Step 2: Square second term $$= 6^2 = 36$$.

    Step 3: Subtract $$= x^2 - 36$$.

    => Correct Answer: A ($$x^2 - 36$$)

  2. Problem 11

    Multiply $$(5x + 2)(5x - 2)$$

    Show solution

    Answer: $$25x^2 - 4$$

    Step 1: First term squared = $$(5x)^2 = 25x^2$$.

    Step 2: Second term squared $$= 2^2 = 4$$.

    Step 3: Subtract second from first $$= 25x^2 - 4$$.

    => Correct Answer: A ($$25x^2 - 4$$)

  3. Problem 12

    Multiply $$(7x^2 - 4y^3)(7x^2 + 4y^3)$$

    Show solution

    Answer: $$49x^4 - 16y^6$$

    Step 1: Square first term = $$(7x^2)^2 = 49x^4$$.

    Step 2: Square second term $$= (4y^3)^2 = 16y^6$$.

    Step 3: Subtract $$= 49x^4 - 16y^6$$.

    => Correct Answer: B ($$49x^4 - 16y^6$$)