Factoring Trinomials of the Form ax² + bx + c: Videos & Practice Problems

Factoring trinomials with a leading coefficient other than one involves methods like trial and error and the AC method. Trial and error uses the FOIL technique to test binomial factors where the first terms multiply to the leading term's coefficient and the last terms multiply to the constant. The AC method rewrites the trinomial into four terms by splitting the middle term based on factors of $a\times c$ that sum to $b$, then applies factoring by grouping. These approaches simplify factoring multivariable polynomials and enhance understanding of polynomial structure and coefficients.

Factoring Trinomials Using Trial and Error

Factoring trinomials with a leading coefficient other than one, such as 2x² or -8x², requires additional strategies beyond simple factoring. One effective approach is the trial and error method, which involves making educated guesses for binomial factors and testing them using the FOIL technique until the original trinomial is reconstructed.

To factor a trinomial like 2x² + 11x + 5, start by identifying pairs of factors for the first term and the last term. The first terms of the binomials must multiply to the leading term, 2x², so possible pairs are x and 2x. The last terms must multiply to the constant term, 5, so possible pairs are 1 and 5. By testing combinations such as (x + 1)(2x + 5) and (x + 5)(2x + 1) using FOIL, you find that (x + 5)(2x + 1) expands back to the original trinomial, confirming the correct factorization.

The FOIL method involves multiplying the First terms, Outside terms, Inside terms, and Last terms of the binomials and then combining like terms. This process ensures the factors multiply correctly to the original trinomial.

For more complex trinomials like 6x² + 19x - 7, begin by listing all factor pairs of the leading coefficient and the constant term, considering their signs carefully. For example, factors of 6x² include 3x and 2x, or x and 6x. Factors of -7 include 7 and -1, or -7 and 1. Then, create all possible binomial pairs from these factors.

Instead of testing every combination with FOIL, focus on the sum of the products of the outside and inside terms, which must equal the middle term, 19x. For example, testing (3x - 1)(2x + 7) involves calculating 3x × 7 = 21x and -1 × 2x = -2x, which sum to 19x, matching the middle term. This confirms the correct factorization.

In summary, factoring trinomials with a leading coefficient other than one involves identifying factor pairs of the first and last terms, then using the FOIL method to test combinations. The key is to find binomial factors whose first terms multiply to ax², last terms multiply to c, and whose outside and inside products sum to the middle term bx. This method enhances problem-solving skills and deepens understanding of polynomial structure.

Key formulas and concepts include:

1. FOIL technique: For binomials \((px + q)(rx + s)\), the product is

\[(px + q)(rx + s) = prx^2 + (ps + qr)x + qs\]

2. Factor pairs must satisfy:

\[pr = a, \quad qs = c, \quad ps + qr = b\]

where \(a\), \(b\), and \(c\) are coefficients from the trinomial \(ax^2 + bx + c\).

By mastering this approach, students can confidently factor a wide range of trinomials, improving algebraic fluency and preparing for more advanced polynomial operations.

Factor the following using trial and error.
$3y^2-14y+8$

• A. $\left(y-2\right)\left(3y-4\right)$
• B. $\left(3y+2\right)\left(y-4\right)$
• C. $\left(3y-2\right)\left(y-4\right)$
• D. $\left(y-4\right)\left(3y+2\right)$

Answer: (C)

Factor the following using trial and error.
$8a^2-22a+15$

• A. $\left(4a-5\right)\left(2a+3\right)$
• B. $\left(4a-5\right)\left(2a-3\right)$
• C. $\left(2a+3\right)\left(4a-5\right)$
• D. $\left(3a-2\right)\left(4a-5\right)$

Answer: (B)

Factoring Trinomials Using Trial and Error Example 1

Factoring trinomials with two variables, such as x and y, follows a similar approach to factoring single-variable trinomials by using the trial and error method. The key is to identify the factors of the first term (the a term) and the last term (the c term) and then test combinations to find binomial pairs that produce the correct middle term when expanded.

For example, if the trinomial has terms involving 2x² and 3y², the factors of a (2) are 1 and 2, and the factors of c (3) are 1 and 3. We then create possible binomial factors such as (1x + 1y) and (2x + 3y), and use the FOIL method to check the middle term. Specifically, we multiply the outside terms: \$1x \times 3y = 3xy\(, and the inside terms: \)1y \times 2x = 2xy\(. Adding these gives \)3xy + 2xy = 5xy\(, which matches the middle term of the trinomial.

This confirms that the trinomial factors as \)(1x + 1y)(2x + 3y)$. This method highlights the importance of understanding how the distributive property and the FOIL technique work together to factor trinomials with multiple variables, reinforcing skills in algebraic manipulation and pattern recognition.

Factoring Trinomials Using Trial and Error Example 2

Factoring trinomials involving two variables, such as x and y, follows a similar approach to single-variable trinomials but requires careful consideration of each term's factors. To factor a trinomial like this, start by identifying the factors of the leading coefficient (the a term) and the constant term (the c term). For example, if the leading coefficient is 2, its factors are 1 and 2; if the constant term is 3, its factors are 1 and 3.

Next, construct possible binomial pairs using these factors, pairing the coefficients with the variables. For instance, one binomial might be 1x + 1y and the other 2x + 3y. To verify if this pair correctly factors the trinomial, apply the FOIL method, focusing on the middle terms since they combine to form the middle term of the trinomial. Multiply the outside terms: \$1x \times 3y = 3xy\(, and the inside terms: \)1y \times 2x = 2xy\(. Adding these gives \)3xy + 2xy = 5xy\(, which matches the middle term of the original trinomial.

Therefore, the trinomial factors as \)(1x + 1y)(2x + 3y)$. This method highlights the importance of trial and error combined with understanding how the FOIL process generates the middle term, enabling the factoring of more complex trinomials involving multiple variables.

Factor completely.
$15a^2+25a-40$

• A. $(3a+8)(a-1)$
• B. $5(3a-8)(a-1)$
• C. $(3a+8)(a+1)$
• D. $5(3a+8)(a-1)$

Answer: (D)

Factoring Trinomials by Grouping (AC Method)

Factoring trinomials can be simplified using the AC method, also known as the grouping method, which is a faster alternative to trial and error. This technique transforms a trinomial with three terms into a four-term polynomial, allowing the use of factoring by grouping. The process begins by identifying the greatest common factor (GCF) of the polynomial; if the GCF is 1, you proceed to the next step.

Next, multiply the leading coefficient a by the constant term c in the trinomial of the form \(ax^2 + bx + c\). Then, find two numbers that multiply to \(a \times c\) and add up to the middle coefficient b. For example, in the trinomial \$2x^2 + 11x + 5\(, multiply \)a = 2\( and \)c = 5\( to get 10. The pair of numbers that multiply to 10 and add to 11 are 1 and 10.

Rewrite the middle term \)11x\( as the sum of these two terms: \)x + 10x\(. This changes the trinomial into a four-term polynomial: \)2x^2 + x + 10x + 5\(. Now, apply factoring by grouping by pairing the first two terms and the last two terms separately. Factor out the GCF from each pair: from \)2x^2 + x\(, factor out \)x\( to get \)x(2x + 1)\(; from \)10x + 5\(, factor out 5 to get \)5(2x + 1)\(. Since both groups contain the common binomial factor \)(2x + 1)\(, factor this out to obtain \)(2x + 1)(x + 5)\(.

This method efficiently factors trinomials where the leading coefficient is not 1 by leveraging the relationship between the coefficients and the distributive property. The key is to find two numbers that satisfy the conditions of multiplying to \)a \times c\( and adding to \)b$, enabling the middle term to be split and the polynomial to be grouped and factored. Mastery of the AC method enhances problem-solving speed and accuracy in algebraic factoring tasks.

Factor completely.
$2x^2+9x+9$

• A. $\left(2x+3\right)\left(x+3\right)$
• B. $\left(2x+3\right)\left(x-3\right)$
• C. $\left(x+3\right)\left(2x-3\right)$
• D. $\left(x-3\right)\left(2x-3\right)$

Answer: (A)

Factor completely.
$-6x^2+17x+3$

• A. $\left(6x+1\right)\left(x+3\right)$
• B. $\left(6x-1\right)\left(-x+3\right)$
• C. $\left(6x+1\right)\left(-x+3\right)$
• D. $\left(6x-1\right)\left(x+3\right)$

Answer: (C)

Factoring Trinomials by Grouping (AC Method) Example 3

To factor a trinomial completely, start by identifying the greatest common factor (GCF) of all terms. In this case, each coefficient is even, indicating a common factor of 2, and every term contains the variable y. Therefore, the GCF is 2y. Factoring out 2y simplifies the trinomial to x² + 5xy + 6y².

Next, focus on factoring the simplified quadratic expression. Since the coefficient of x² is 1, look for two numbers that multiply to the constant term (6y²) and add up to the middle coefficient (5y). The pairs to consider are (1y, 6y) and (2y, 3y). The pair (2y, 3y) works because 2y + 3y = 5y and 2y × 3y = 6y².

Thus, the trinomial factors as (x + 2y)(x + 3y). Incorporating the GCF back, the fully factored form is 2y(x + 2y)(x + 3y). This process highlights the importance of factoring out common terms first to simplify complex expressions before applying standard factoring techniques.

Factor the following polynomial
$6x^3+9x^2-15x$

• A. $3x\left(x+1\right)\left(2x-5\right)$
• B. $3x\left(x-1\right)\left(2x+5\right)$
• C. $3x\left(2x-5\right)\left(x-1\right)$
• D. $\left(x-1\right)\left(2x+5\right)$

Answer: (B)

Factor the following polynomial
$2x^2+7xy+3y^2$

• A. $\left(x+2y\right)\left(3x+y\right)$
• B. $\left(2x+y\right)\left(x+3y\right)$
• C. $\left(2x-y\right)\left(x+3y\right)$
• D. $\left(x+3y\right)\left(2x+y\right)$

Answer: (B)

Here’s what students ask on this topic:

How do you factor trinomials when the leading coefficient is not 1?

When the leading coefficient $a$ in a trinomial $\mathrm{ax}{x}^2+\mathrm{bx}+c$ is not 1, you can use methods like trial and error or the AC method. Trial and error involves guessing binomial factors where the first terms multiply to $a{x}^2$ and the last terms multiply to $c$. You then use the FOIL technique to check if the product matches the original trinomial. The AC method involves multiplying $a$ and $c$, finding two numbers that multiply to $a\times c$ and add to $b$, splitting the middle term accordingly, and then factoring by grouping. Both methods help simplify factoring trinomials with leading coefficients other than one.

What is the trial and error method for factoring trinomials?

The trial and error method for factoring trinomials involves making educated guesses for binomial factors. For a trinomial $\mathrm{ax}{x}^2+\mathrm{bx}+c$, you first find pairs of terms whose product equals the first term $\mathrm{ax}{x}^2$ and the last term $c$. Then, you test different combinations of these pairs using the FOIL method to see if their product equals the original trinomial. This process continues until the correct binomial factors are found. Although it can be time-consuming, it is a straightforward way to factor trinomials when the leading coefficient is not one.

How does the AC method simplify factoring trinomials?

The AC method simplifies factoring trinomials by converting a three-term polynomial into four terms, making it easier to factor by grouping. For a trinomial $\mathrm{ax}{x}^2+\mathrm{bx}+c$, you multiply $a$ and $c$ to find their product. Then, find two numbers that multiply to $a\times c$ and add to $b$. These numbers split the middle term into two terms. After rewriting the trinomial with four terms, you factor by grouping, extracting common factors from pairs of terms. This method is often faster and more systematic than trial and error, especially for trinomials with larger coefficients.

What are the steps to factor a trinomial using the AC method?

To factor a trinomial $\mathrm{ax}{x}^2+\mathrm{bx}+c$ using the AC method, follow these steps: (1) Multiply $a$ and $c$. (2) Find two numbers that multiply to $a\times c$ and add to $b$. (3) Rewrite the middle term $\mathrm{bx}$ as the sum of two terms using these numbers. (4) Group the four terms into two pairs. (5) Factor out the greatest common factor from each pair. (6) Factor out the common binomial factor. This results in the factored form of the original trinomial.

How do you use the FOIL method to check your factors?

The FOIL method helps verify if your binomial factors multiply back to the original trinomial. FOIL stands for First, Outside, Inside, Last, referring to the terms you multiply: (1) Multiply the first terms of each binomial. (2) Multiply the outside terms. (3) Multiply the inside terms. (4) Multiply the last terms. Then, add all these products together. If the sum matches the original trinomial, your factors are correct. This method is essential when using trial and error to test possible binomial factors.