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 term. 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.

When factoring a trinomial of the form ax² + bx + c, the first step is to identify pairs of factors for the first term ax² and the last term c. For example, in the trinomial 2x² + 11x + 5, the factors of 2x² could be x and 2x, while the factors of 5 are 1 and 5. Possible binomial factors are then formed by pairing these factors, such as (x + 1)(2x + 5) or (x + 5)(2x + 1). Using the FOIL method, which stands for First, Outside, Inside, Last, you multiply the terms in each binomial pair and combine like terms to check if the product matches the original trinomial. The correct factorization is the pair that, when multiplied, yields the original trinomial exactly.

For more complex trinomials like 6x² + 19x - 7, the process involves listing all factor pairs of 6x² (such as 3x and 2x, or x and 6x) and all factor pairs of the constant term, considering their signs carefully (since the constant is negative, factors like 7 and -1 or -7 and 1 are possible). Then, all possible binomial combinations are tested. However, instead of fully expanding each pair, focus on the sum of the products of the outside and inside terms from FOIL, which must equal the middle term coefficient (in this case, 19x). This shortcut helps quickly eliminate incorrect pairs.

For instance, 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. Always verify by multiplying the binomials fully to ensure the original trinomial is obtained.

In summary, factoring trinomials with a leading coefficient other than one involves identifying factor pairs of the first and last terms, forming binomial pairs, and using the FOIL method to test these pairs. Paying close attention to the sum of the inner and outer products streamlines the process, making trial and error more efficient and effective.

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) in the trinomial. For example, if the first term is 2x², its factors could be 1x and 2x. If the last term is 3y², its factors could be 1y and 3y.

Next, you create possible binomial pairs by combining these factors. For instance, pairing 1x with 2x and 1y with 3y gives potential binomials like (1x + 1y) and (2x + 3y). To verify if these binomials correctly factor the trinomial, focus on the middle terms generated by the FOIL method, specifically the sum of the products of the inside and outside terms. Multiplying the outside terms, 1x × 3y, results in 3xy, and multiplying the inside terms, 1y × 2x, results in 2xy. Adding these together yields 5xy, which matches the middle term of the original trinomial.

Therefore, the trinomial can be factored as \((x + y)(2x + 3y)\). This method highlights the importance of understanding how to factor trinomials with multiple variables by systematically testing factor pairs and verifying the middle term through partial FOIL multiplication.

Factoring Trinomials Using Trial and Error Example 2

To factor a trinomial completely, it is effective to first identify and factor out the greatest common factor (GCF). This simplifies the polynomial and makes subsequent factoring easier, especially when dealing with large coefficients. For example, consider the trinomial with terms 8x², 28x, and 16. By expressing each term in its prime factorization, we find:

• 8x² = 2 × 2 × 2 × x × x
• 28x = 2 × 2 × 7 × x
• 16 = 2 × 2 × 2 × 2

The common factors among all terms include at least two 2's, so the GCF is 4. Factoring out 4 from the trinomial yields:

\$4(2x^2 - 7x - 4)\(

Next, the trinomial inside the parentheses can be factored using the trial and error method. This involves finding factor pairs of the leading coefficient (a = 2) and the constant term (c = -4). The factors of 2 are 2 and 1, while the factors of -4 include (-4, 1), (4, -1), (2, -2), and (-2, 2).

By pairing these factors and focusing on the middle term, which comes from the sum of the products of the outer and inner terms in the FOIL method, we test combinations. For instance, pairing 2x and 1x with 1 and -4, we calculate:

Outside terms: \)2x \times (-4) = -8x\(

Inside terms: \)1 \times 1x = 1x\(

Adding these gives \)-8x + 1x = -7x\(, which matches the middle term's coefficient.

Thus, the trinomial factors as:

\)4(2x + 1)(x - 4)$

This process highlights the importance of factoring out the GCF first to simplify the polynomial, then systematically testing factor pairs to factor the trinomial completely.

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 often 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 three-term polynomial into four terms, 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 simultaneously add up to the middle coefficient b. For example, in the trinomial \$2x^2 + 11x + 5\(, multiplying \)a = 2\( and \)c = 5\( gives 10. The pair of numbers that multiply to 10 and add to 11 are 1 and 10.

After identifying these numbers, rewrite the middle term \)bx\( as the sum of two terms using these numbers: \)11x = x + 10x\(. This converts the trinomial into four terms: \)2x^2 + x + 10x + 5\(. Then, apply factoring by grouping by pairing the first two terms and the last two terms separately. Factor out the GCF from each pair. For the first pair, \)2x^2 + x\(, the GCF is \)x\(, resulting in \)x(2x + 1)\(. For the second pair, \)10x + 5\(, the GCF is 5, resulting in \)5(2x + 1)\(.

Since both groups contain the common binomial factor \)(2x + 1)\(, factor this out to get the final factored form: \)(2x + 1)(x + 5)\(. This method is efficient and reliable for factoring trinomials where the leading coefficient is not 1, as long as you can find two numbers that satisfy the multiplication and addition conditions related to \)a \times c\( and \)b$.

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) among all terms. In this case, each coefficient is even, indicating a common factor of 2, and every term contains the variable y, so 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 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 first extracting the greatest common factor to simplify complex expressions before applying factoring techniques such as finding two numbers that multiply to the constant term and add to the middle coefficient. Factoring trinomials with variables and coefficients becomes manageable by breaking down the problem step-by-step.

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 $\mathrm{ax}{x}^2+bx+c$ is not 1, you can use methods like trial and error or the AC method. Trial and error involves guessing binomial factors whose first terms multiply to $a{x}^2$ and last terms multiply to $c$. You test these guesses using the FOIL technique until you find the correct factors. The AC method multiplies $a$ and $c$, finds two numbers that multiply to $a\times c$ and add to $b$, then splits the middle term accordingly. After rewriting the trinomial as four terms, you factor by grouping. Both methods help simplify factoring trinomials with leading coefficients other than 1.

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 $\mathrm{ax}{x}^2+bx+c$, you first find pairs of terms whose product equals the first term $a{x}^2$ and pairs whose product equals the last term $c$. You then combine these pairs into binomials and use the FOIL technique to multiply them. If the result matches the original trinomial, you have found the correct factors. If not, you try different combinations until you find the right one. This method relies on pattern recognition and testing possible factor pairs systematically.

How does the AC method simplify factoring trinomials with a leading coefficient other than one?

The AC method simplifies factoring trinomials $\mathrm{ax}{x}^2+bx+c$ by first multiplying $a$ and $c$ to get a product. Then, you find two numbers that multiply to this product and add to $b$. These two numbers are used to split the middle term into two terms, rewriting the trinomial as four terms. Next, you factor by grouping, pairing terms and factoring out common factors. This method reduces the trial and error process by focusing on the product $a\times c$ and the sum $b$, making it faster and more systematic to factor trinomials with leading coefficients other than one.

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

To factor a trinomial $\mathrm{ax}{x}^2+bx+c$ 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 $bx$ 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 process transforms the trinomial into a product of two binomials, making factoring more efficient.

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

The FOIL technique 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 equals the original trinomial $\mathrm{ax}{x}^2+bx+c$, your factors are correct. This method confirms the accuracy of your factorization.