Factoring Trinomials of the Form x² + bx + c: Videos & Practice Problems

Factoring a trinomial of the form $x^2+bx+c$ involves finding two binomials whose product equals the original polynomial. Using the FOIL method, the constant term $c$ is the product of two constants, and the coefficient $b$ is their sum. For example, to factor $x^2\; +\; 10x\; +\; 21$, find two numbers multiplying to 21 and adding to 10, which are 3 and 7, resulting in $(x\; +\; 3)(x\; +\; 7)$. This technique applies when the coefficient of $x^2$ is 1, emphasizing understanding polynomial terms, coefficients, and factoring strategies.

Factoring Trinomials of the Form x² + bx + c

Factoring trinomials of the form x² + bx + c is a fundamental skill in algebra that allows you to rewrite quadratic expressions as the product of two binomials. This process essentially reverses the distributive property or FOIL (First, Outer, Inner, Last) method used to multiply binomials. For example, multiplying (x + 3)(x + 7) using FOIL results in x² + 10x + 21. Factoring works by identifying two numbers that both multiply to the constant term c and add up to the coefficient b of the middle term.

To factor a trinomial like x² + 10x + 21, you look for two numbers that multiply to 21 and add to 10. These numbers are 3 and 7, so the factored form is (x + 3)(x + 7). This approach relies on recognizing patterns from the FOIL method: the product of the last terms gives c, and the sum of the outer and inner terms gives b.

When the numbers are less straightforward, a systematic approach helps. For instance, to factor x² + 3x − 28, identify b = 3 and c = −28. List factor pairs of −28 and check which pair sums to 3. Using a T-chart to organize factor pairs such as (−4, 7) and (4, −7) makes this easier. Since −4 + 7 = 3, the trinomial factors as (x − 4)(x + 7).

Similarly, for x² − 11x + 30, with b = −11 and c = 30, factor pairs of 30 include (1, 30), (2, 15), (3, 10), and (5, 6). Since −5 + (−6) = −11 and (−5)(−6) = 30, the trinomial factors as (x − 5)(x − 6).

This factoring method only applies when the coefficient of x² is 1. To verify your factorization, multiply the binomials back using FOIL to ensure you return to the original trinomial. Mastering this technique enhances your ability to solve quadratic equations and simplifies expressions efficiently.

Factor the following trinomials completely.
$y^2-7y+12$

• A. $\left(y+3\right)\left(y-4\right)$
• B. $\left(y-3\right)\left(y-4\right)$
• C. $\left(y+7\right)\left(y-1\right)$
• D. $\left(y-1\right)\left(y-12\right)$

Answer: (B)

Factor the following trinomials completely.
$z^2-11z+30$

• A. $\left(z+5\right)\left(z-6\right)$
• B. $\left(z-3\right)\left(z-10\right)$
• C. $\left(z+5\right)\left(z+6\right)$
• D. $\left(z-5\right)\left(z-6\right)$

Answer: (D)

Factoring Trinomials of the Form x² + bx + c Example 1

When factoring a trinomial with a leading term of x², the method involves finding two numbers that multiply to the constant term (the c value) and add to the coefficient of the middle term (the b value). This approach remains effective even when the trinomial contains multiple variables, such as x and y.

Consider a trinomial like x² - 5xy + 6y². To factor it, first identify pairs of factors of the constant term, which here is 6y². Possible factor pairs include (1y, 6y) and (2y, 3y). Since the middle term is -5xy, the two numbers must add to -5y. The pair -2y and -3y satisfies this because (-2y) + (-3y) = -5y, and their product is 6y².

Thus, the trinomial factors as:

\[(x - 2y)(x - 3y)\]

To verify, apply the FOIL method (First, Outer, Inner, Last):

• First: \(x \times x = x^2\)
• Outer: \(x \times (-3y) = -3xy\)
• Inner: \((-2y) \times x = -2xy\)
• Last: \((-2y) \times (-3y) = 6y^2\)

Combining like terms yields:

\[x^2 - 3xy - 2xy + 6y^2 = x^2 - 5xy + 6y^2\]

This confirms the factorization is correct. Understanding how to factor trinomials with multiple variables by identifying appropriate factor pairs and verifying through expansion is essential for mastering polynomial factorization.

Here’s what students ask on this topic:

How do you factor a trinomial of the form $x^2+bx+c$?

To factor a trinomial of the form $x^2+bx+c$, you need to find two numbers that multiply to $c$ and add to $b$. These two numbers will be the constants in the binomials that multiply to give the original trinomial. The factored form will be $(x+p)(x+q)$, where $p$ and $q$ satisfy $pimesq=c$ and $p+q=b$. This method reverses the FOIL process used in multiplying binomials.

What is the systematic approach to factoring trinomials when mental math is difficult?

When mental math is challenging, use a systematic approach by listing all factor pairs of $c$ in a T-chart. For each pair, check if their sum equals $b$. Remember to consider both positive and negative factors, especially if $c$ or $b$ is negative. Once you find the pair that adds to $b$, write the factored form as $(x+p)(x+q)$. This method helps organize your work and reduces errors.

Can you factor trinomials of the form $x^2+\; bx\; +\; c$ when the coefficient of $x^2$ is not 1?

The method described applies only when the coefficient of $x^2$ is 1. If the coefficient is not 1, such as in $ax^2\; +\; bx\; +\; c$ where $a\; eq\; 1$, you need to use other factoring techniques like factoring by grouping or the AC method. These methods account for the leading coefficient and are more complex. Always check the coefficient of $x^2$ before applying the simple factoring method.

How do you factor $x^2+\; 3x\; -\; 28$ using the factoring method?

First, identify $b\; =\; 3$ and $c\; =\; -28$. List factor pairs of $-28$: (1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), (-4, 7). Check which pair sums to 3. The pair $-4$ and $7$ works because $-4\; +\; 7\; =\; 3$. Therefore, the factored form is $(x-4)(x+7)$.

How can you verify your factoring of a trinomial?

To verify your factoring, multiply the binomials back using the FOIL method: multiply the First terms, Outside terms, Inside terms, and Last terms, then combine like terms. If the product matches the original trinomial, your factoring is correct. For example, if you factored $x^2\; +\; 10x\; +\; 21$ as $(x\; +\; 3)(x\; +\; 7)$, multiply to get $x^2\; +\; 7x\; +\; 3x\; +\; 21\; =\; x^2\; +\; 10x\; +\; 21$, confirming the factorization.