How to Factor Using Addition and Subtraction?

Factoring is a crucial skill in algebra that involves breaking down an expression into simpler components, or ‘factors,’ which, when multiplied together, give the original expression. Factoring using addition and subtraction specifically refers to methods that involve manipulating the terms of an expression to reveal its factors.

Basic Concepts

What is Factoring?

Factoring is the process of expressing a mathematical expression as a product of its factors. For example, factoring the number 12 gives us $12 = 2 times 6$ or $12 = 3 times 4$. In algebra, we apply similar principles to polynomials and other algebraic expressions.

Why Use Addition and Subtraction?

Using addition and subtraction in factoring helps in rearranging and simplifying terms to make the factoring process easier. This method is especially useful for quadratic expressions and polynomials.

Factoring Quadratic Expressions

Quadratic expressions are of the form $ax^2 + bx + c$. One common method of factoring such expressions involves finding two numbers that add up to $b$ (the coefficient of $x$) and multiply to $ac$ (the product of the coefficient of $x^2$ and the constant term).

Example 1: Factoring $x^2 + 5x + 6$

  1. Identify $a$, $b$, and $c$:
    • Here, $a = 1$, $b = 5$, and $c = 6$
  2. Find two numbers that add to $b$ and multiply to $ac$:
    • We need two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3.
  3. Rewrite the middle term using these numbers:
    • $x^2 + 5x + 6 = x^2 + 2x + 3x + 6$
  4. Factor by grouping:
    • Group the terms: $(x^2 + 2x) + (3x + 6)$
    • Factor out the common factors: $x(x + 2) + 3(x + 2)$
    • Factor out the common binomial: $(x + 2)(x + 3)$

So, $x^2 + 5x + 6$ factors to $(x + 2)(x + 3)$

Example 2: Factoring $2x^2 + 7x + 3$

  1. Identify $a$, $b$, and $c$:
    • Here, $a = 2$, $b = 7$, and $c = 3$
  2. Find two numbers that add to $b$ and multiply to $ac$:
    • We need two numbers that add up to 7 and multiply to 6 (2 * 3). These numbers are 1 and 6.
  3. Rewrite the middle term using these numbers:
    • $2x^2 + 7x + 3 = 2x^2 + x + 6x + 3$
  4. Factor by grouping:
    • Group the terms: $(2x^2 + x) + (6x + 3)$
    • Factor out the common factors: $x(2x + 1) + 3(2x + 1)$
    • Factor out the common binomial: $(2x + 1)(x + 3)$

So, $2x^2 + 7x + 3$ factors to $(2x + 1)(x + 3)$

Factoring by Completing the Square

Another method that involves addition and subtraction is completing the square. This method is particularly useful for solving quadratic equations and can also be used for factoring.

Example: Factoring $x^2 + 6x + 9$

  1. Rewrite the expression:
    • $x^2 + 6x + 9$
  2. Identify the coefficient of $x$ and divide by 2, then square it:
    • The coefficient of $x$ is 6. Dividing by 2 gives 3, and squaring it gives 9.
  3. Rewrite the expression as a perfect square trinomial:
    • $x^2 + 6x + 9 = (x + 3)^2$

So, $x^2 + 6x + 9$ factors to $(x + 3)^2$

Factoring Polynomials

For polynomials of higher degrees, similar principles apply. The goal is to rearrange terms using addition and subtraction to reveal common factors or to simplify the expression into a product of simpler polynomials.

Example: Factoring $x^3 + 3x^2 – 4x – 12$

  1. Group terms to identify common factors:
    • $(x^3 + 3x^2) + (-4x – 12)$
  2. Factor out the common factors in each group:
    • $x^2(x + 3) – 4(x + 3)$
  3. Factor out the common binomial factor:
    • $(x^2 – 4)(x + 3)$
  4. Factor further if possible:
    • $(x^2 – 4)$ can be factored as a difference of squares: $(x + 2)(x – 2)$

So, $x^3 + 3x^2 – 4x – 12$ factors to $(x + 2)(x – 2)(x + 3)$

Practice Problems

  1. Factor $x^2 + 8x + 15$
  2. Factor $3x^2 + 11x + 6$
  3. Factor $x^3 – 2x^2 – 9x + 18$

Conclusion

Factoring using addition and subtraction is a versatile and essential technique in algebra. By understanding and practicing these methods, you can simplify complex expressions and solve equations more efficiently. Whether you’re dealing with quadratics, completing the square, or higher-degree polynomials, these strategies will serve you well in your mathematical journey.

Citations

  1. 1. Khan Academy – Factoring quadratics
  2. 2. Purplemath – Factoring Polynomials
  3. 3. Math is Fun – Factoring

Related

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H + HO2 → O2 + H2 k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O2 k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) H + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-5 s^-1) φ

Table 1 Reactions, rate constants and activation energies used in the model* No. Reaction kopt (M⁻¹ s⁻¹) 1 OH + H₂ → H + H₂O 3.74 x 10⁷ 2 OH + HO₂ → HO₂ + OH⁻ 5 x 10⁹ 3 OH + H₂O₂ → HO₂ + H₂O 3.8 x 10⁷ 4 OH + O₂ → O₂ + OH 9.96 x 10⁹ 5 OH + HO₂ → O₂ + H₂O 7.1 x 10⁹ 6 OH + OH → H₂O₂ 5.3 x 10⁹ 7 OH + e⁻aq → OH⁻ 3 x 10¹⁰ 8 H + O₂ → HO₂ 2.0 x 10¹⁰ 9 H + HO₂ → H₂O₂ 2.0 x 10¹⁰ 10 H + H₂O₂ → OH + H₂O 3.44 x 10⁷ 11 H + OH → H₂O 1.4 x 10¹⁰ 12 H + H → H₂ 1.94 x 10¹⁰ 13 e⁻aq + O₂ → O₂⁻ 1.9 x 10¹⁰ 14 e⁻aq + O₂ → HO₂⁻ + OH⁻ 1.3 x 10¹⁰ 15 e⁻aq + HO₂ 2.0 x 10¹⁰ 16 e⁻aq + H₂O₂ 1.1 x 10¹⁰ 17 e⁻aq + HO₂ → OH + OH⁻ 1.3 x 10¹⁰ 18 e⁻aq + H⁺ → H 2.3 x 10¹⁰ 19 e⁻aq + e⁻aq → H₂ + OH⁻ + OH⁻ 2.5 x 10⁹ 20 HO₂ + O₂ → O₂ + HO₂ 1.3 x 10⁹ 21 HO₂ + HO₂ → O₂ + H₂O₂ 8.3 x 10⁵ 22 HO₂ + HO₂ → O₂ + OH + H₂O 3.7 23 HO₂ + HO₂ → O₂ + O₂ + OH + H₂O 7 x 10⁵ s⁻¹ 24 H⁺ + O₂⁻ → HO₂ 4.5 x 10¹⁰ 25 H⁺ + O₂⁻ → O₂ 2.0 x 10¹⁰ 26 H⁺ + OH⁻ 1.4 x 10¹¹ 27 H⁺ + HO₂⁻ 2 x 10¹⁰ 28 H₂O₂ → HO₂ + H⁺ + OH⁻ 2.5 x 10⁻⁵ s⁻¹ 29 H₂O₂ → H⁺ + OH⁻ 1.4 x 10⁻⁷ s⁻¹ 30 O₂ + O₂ → O₂ + HO₂ + OH⁻ 0.3 31 O₂ + H₂O₂ → O₂ + OH + OH 16 32

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H2O + O → 2 OH k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) OH + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-8 s^-1) φ