7.1 Polynomial Operations & Factoring Techniques

7.1 Polynomial Operations & Factoring Techniques

Polynomials are fundamental algebraic expressions encountered throughout the Armed Services Vocational Aptitude Battery (ASVAB) Mathematics Knowledge (MK) subtest. Understanding how to classify, add, subtract, multiply, and factor polynomials is essential for scoring in the higher percentiles and qualifying for technical military occupational specialties.

1. What is a Polynomial?

A polynomial is an algebraic expression composed of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication.

Key Terminology

• Monomial: A polynomial with a single term (e.g., $5x^3$, $-7$, $12xy^2$).
• Binomial: A polynomial with exactly two terms connected by addition or subtraction (e.g., $3x + 4$, $x^2 - 25$).
• Trinomial: A polynomial with exactly three terms (e.g., $x^2 - 5x + 6$, $2a^2 + 3ab - b^2$).
• Degree of a Polynomial: The highest exponent of the variable in a single-variable polynomial. For instance, in $4x^3 - 2x^2 + 7x - 1$, the degree is 3 (a cubic polynomial). In $x^2 + 5x - 6$, the degree is 2 (a quadratic polynomial).
• Leading Coefficient: The numerical coefficient of the term with the highest degree. In $4x^3 - 2x^2 + 7x - 1$, the leading coefficient is 4.
• Standard Form: A polynomial written with terms arranged in descending order of degree (e.g., $3x^3 - 5x^2 + 2x + 9$).

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2. Addition and Subtraction of Polynomials

Adding and subtracting polynomials involves identifying and combining like terms—terms that share the exact same variable(s) raised to the exact same power(s).

Adding Polynomials

To add two or more polynomials, remove parentheses and group like terms together.

Worked Example 1: Addition

Simplify $(4x^2 - 3x + 7) + (2x^2 + 8x - 5)$.

1. Group like terms: $(4x^2 + 2x^2) + (-3x + 8x) + (7 - 5)$
2. Combine coefficients: $6x^2 + 5x + 2$

Subtracting Polynomials

Subtracting polynomials requires extra care. You must distribute the negative sign (multiply by $-1$) across every term inside the second set of parentheses before combining like terms.

Crucial Rule: $(A - B) - (C - D) = A - B - C + D$. A common error on the AFQT is forgetting to flip the sign of the constant or second term inside the subtracted expression.

Worked Example 2: Subtraction

Simplify $(5x^2 - 4x + 9) - (2x^2 - 7x - 3)$.

1. Distribute the negative sign across the second polynomial: $5x^2 - 4x + 9 - 2x^2 + 7x + 3$
2. Group like terms: $(5x^2 - 2x^2) + (-4x + 7x) + (9 + 3)$
3. Combine terms: $3x^2 + 3x + 12$

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3. Multiplying Polynomials

Polynomial multiplication relies heavily on the distributive property ($a(b + c) = ab + ac$) and the exponent rule for multiplication ($x^a \cdot x^b = x^{a+b}$).

Monomial by Polynomial

Multiply the outside term by every term inside the polynomial.

Example:

$3x^2(4x^2 - 5x + 2) = (3x^2 \cdot 4x^2) - (3x^2 \cdot 5x) + (3x^2 \cdot 2) = 12x^4 - 15x^3 + 6x^2$

Binomial by Binomial: The FOIL Method

When multiplying two binomials $(a + b)(c + d)$, use the FOIL acronym to ensure all four products are calculated:

Summing the results: $2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$

Worked Example 3: FOIL Expansion

Expand and simplify $(3x - 4)(2x + 7)$.

1. First: $3x \cdot 2x = 6x^2$
2. Outer: $3x \cdot 7 = 21x$
3. Inner: $-4 \cdot 2x = -8x$
4. Last: $-4 \cdot 7 = -28$
5. Combine Inner and Outer terms: $21x - 8x = 13x$
6. Final Expression: $6x^2 + 13x - 28$

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4. Factoring Polynomials

Factoring is the process of reversing polynomial multiplication—turning an expression into a product of simpler terms. Factoring is a primary method for solving equations, simplifying algebraic fractions, and analyzing functions on the AFQT.

Step 1: Factoring Out the Greatest Common Factor (GCF)

Always start any factoring problem by looking for the largest factor common to all terms.

Worked Example 4: GCF Factoring

Factor $12x^3y^2 - 18x^2y^3 + 6x^2y$.

1. Identify numerical GCF: The greatest common factor of $12, 18, 6$ is $6$.
2. Identify variable GCF:
   • For $x$: powers are $x^3, x^2, x^2 \rightarrow$ GCF is $x^2$.
   • For $y$: powers are $y^2, y^3, y^1 \rightarrow$ GCF is $y$.
3. Overall GCF: $6x^2y$.
4. Divide each term by the GCF: $6x^2y \left( \frac{12x^3y^2}{6x^2y} - \frac{18x^2y^3}{6x^2y} + \frac{6x^2y}{6x^2y} \right) = 6x^2y(2xy - 3y^2 + 1)$

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5. Special Factoring Patterns

Recognizing algebraic patterns allows you to factor complex expressions instantly without tedious trial and error.

Pattern 1: Difference of Two Squares

Any binomial in the form $a^2 - b^2$ factors into conjugate pairs: $a^2 - b^2 = (a - b)(a + b)$

Warning: The sum of two squares ($a^2 + b^2$) cannot be factored over real numbers!

Examples of Difference of Squares:

• $x^2 - 49 = (x - 7)(x + 7)$
• $16x^2 - 25 = (4x - 5)(4x + 5)$
• $81a^2 - 100b^2 = (9a - 10b)(9a + 10b)$

Pattern 2: Perfect Square Trinomials

When a trinomial is produced by squaring a binomial, it fits one of two standard forms: $a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

To test if a trinomial $x^2 + bx + c$ is a perfect square:

1. Check if the first and last terms are perfect squares: $\sqrt{x^2} = x$ and $\sqrt{c} = \sqrt{c}$.
2. Check if the middle term $b$ equals $2 \cdot \sqrt{x^2} \cdot \sqrt{c}$.

Examples of Perfect Square Trinomials:

• $x^2 + 10x + 25$: Since $\sqrt{25} = 5$ and $2 \cdot 5 = 10$, this factors as $(x + 5)^2$.
• $x^2 - 12x + 36$: Since $\sqrt{36} = 6$ and $2 \cdot 6 = 12$, this factors as $(x - 6)^2$.
• $4x^2 + 20x + 25$: Since $\sqrt{4x^2} = 2x$ and $\sqrt{25} = 5$, and $2(2x)(5) = 20x$, this factors as $(2x + 5)^2$.

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6. Factoring Standard Trinomials ($x^2 + bx + c$)

To factor a quadratic trinomial where the leading coefficient is 1 ($x^2 + bx + c$), find two numbers $p$ and $q$ such that: $p \cdot q = c \quad \text{and} \quad p + q = b$ Once found, the factored form is $(x + p)(x + q)$.

Signs Guide for Trinomial Factoring

Worked Example 5: Factoring $x^2 - 5x - 24$

1. $c = -24$, so one factor is positive and one is negative.
2. $b = -5$, so the larger factor in absolute value must be negative.
3. Factor pairs of $24$: $(1, 24), (2, 12), (3, 8), (4, 6)$.
4. Select the pair with a difference of $5$: $+3$ and $-8$ because $3 \cdot (-8) = -24$ and $3 + (-8) = -5$.
5. Factored form: $(x - 8)(x + 3)$.

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7. AFQT Strategies, Tips & Pitfalls

• Always Check for GCF First: Before attempting FOIL factoring or quadratic formulas, factor out common numerical or variable factors. For example, $3x^2 - 12x - 36 = 3(x^2 - 4x - 12) = 3(x - 6)(x + 2)$.
• Beware of the Subtraction Sign Trap: When subtracting polynomials like $(2x^2 + 5x) - (x^2 - 3x)$, ensure you convert $-(-3x)$ to $+3x$. The correct result is $x^2 + 8x$, not $x^2 + 2x$.
• Test Options by Expanding (Back-solving): If you are unsure how to factor a trinomial on the test, multiply the answer choices using FOIL to see which option expands back to the original question!