Factor the following trinomials completely.y2−$$7y+12y^2-7y+12$$

(y−3)(y−4)\left(y-3\right)\left(y-4\right)(y−3)(y−4)

(y+3)(y−4)\left(y+3\right)\left(y-4\right)(y+3)(y−4)

(y+7)(y−1)\left(y+7\right)\left(y-1\right)(y+7)(y−1)

(y−1)(y−12)\left(y-1\right)\left(y-12\right)(y−1)(y−12)

Factor the following trinomials completely.z2−$$11z+30z^2-11z+30$$

(z−5)(z−6)\left(z-5\right)\left(z-6\right)(z−5)(z−6)

(z+5)(z−6)\left(z+5\right)\left(z-6\right)(z+5)(z−6)

(z−3)(z−10)\left(z-3\right)\left(z-10\right)(z−3)(z−10)

(z+5)(z+6)\left(z+5\right)\left(z+6\right)(z+5)(z+6)

7. Factoring

Factoring Trinomials of the Form x² + bx + c

7. Factoring

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

Video Lessons Worksheet

Topic summary

Factoring a trinomial of the form x2+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. By identifying factor pairs of c that add to b, students can rewrite the trinomial as (x + p)(x + q). This technique applies when the coefficient of x^2 is one, reinforcing understanding of polynomials, coefficients, and binomials.

Concept

Factoring Trinomials of the Form x² + bx + c

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Factoring Trinomials of the Form x² + bx + c Video Summary

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.

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Problem

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

$$\left$(y+3$\right$)$\left$(y-4$\right$)$

$$\left$(y-3$\right$)$\left$(y-4$\right$)$

$$\left$(y+7$\right$)$\left$(y-1$\right$)$

$$\left$(y-1$\right$)$\left$(y-12$\right$)$

Problem

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

$$\left$(z+5$\right$)$\left$(z-6$\right$)$

$$\left$(z-3$\right$)$\left$(z-10$\right$)$

$$\left$(z+5$\right$)$\left$(z+6$\right$)$

$$\left$(z-5$\right$)$\left$(z-6$\right$)$

Example

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

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

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:

To factor a trinomial of the form $x^{2} + b x + c$ , you need to find two numbers that multiply to $c$ and add to $b$ . These two numbers will be the constants in the binomials. The trinomial can then be factored as $(x + p) (x + q)$ , where $p$ and $q$ are the numbers found. This method works only when the coefficient of $x^{2}$ is 1. For example, to factor $x^{2} + 10 x + 21$ , find two numbers that multiply to 21 and add to 10, which are 3 and 7. So, the factorization is $(x + 3) (x + 7)$ .

The FOIL method helps understand how to factor trinomials by working backward from multiplication. FOIL stands for First, Outside, Inside, Last, which are the steps used to multiply two binomials. When you multiply $(x + p) (x + q)$ , FOIL gives $x^{2} + (p+q)x + pq$ . Factoring reverses this process: given $x^{2} + bx + c$ , you find $p$ and $q$ such that $p + q = b$ and $pq = c$ . This understanding is crucial because it connects multiplication and factoring, allowing you to identify the binomials that produce the original trinomial.

To find two numbers that multiply to $c$ and add to $b$ , start by listing all factor pairs of $c$ . For example, if $c = 28$ , the factor pairs are (1, 28), (2, 14), (4, 7), and their negative counterparts. Then, check which pair sums to $b$ . If $b$ is positive, both numbers are positive or both negative if $c$ is positive. If $b$ is negative, one or both numbers will be negative accordingly. Using a T-chart can help organize these pairs and sums systematically. Once you find the correct pair, you use them to write the factored form as $(x + p) (x + q)$ .

The method described for factoring trinomials of the form $x^2 + bx + c$ works only when the coefficient $a$ of $x^2$ is 1. When $a$ is not 1, the process is more complex because the product of the first terms in the binomials is no longer just $x^2$ . In such cases, other methods like factoring by grouping or using the AC method are used. These methods involve multiplying $a$ and $c$ , finding factors of that product that add to $b$ , and then grouping terms to factor stepwise. So, the simple approach of finding two numbers that multiply to $c$ and add to $b$ applies only when $a = 1$ .

To check if your factorization is correct, multiply the binomials back together using the FOIL method. For example, if you factored $x^2 + 10x + 21$ as $(x + 3)(x + 7)$ , multiply: First: $x imes x = x^2$ , Outside: $x imes 7 = 7x$ , Inside: $3 imes x = 3x$ , Last: $3 imes 7 = 21$ . Adding the middle terms gives $7x + 3x = 10x$ . The product is $x^2 + 10x + 21$ , which matches the original trinomial. If the product matches, your factorization is correct. This step is important to confirm your solution and avoid errors.