7. Factoring

Factoring Trinomials of the Form x² + bx + c

7. Factoring

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

Topic summary

Factoring a trinomial of the form $x^{2} + b x + c$ involves finding two binomials whose product equals the trinomial. 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 factor the polynomial into $(x + p)(x + q)$ . 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, Outside, Inside, Last) method used to multiply binomials. When multiplying two binomials such as (x + p)(x + q), the result is x² + (p + q)x + pq. Recognizing this pattern is key to factoring.

To factor a trinomial x² + bx + c, you need to find two numbers that multiply to c and add to b. These two numbers become the constants in the binomials (x + p)(x + q). For example, factoring x² + 10x + 21 involves finding two numbers that multiply to 21 and add to 10, which are 3 and 7. Thus, the factorization is (x + 3)(x + 7).

When the numbers are less straightforward, a systematic approach helps. Start by identifying the values of b and c. Then list all factor pairs of c, considering both positive and negative pairs depending on the signs of b and c. Use a T-chart or organized list to check which pair sums to b. For instance, to factor x² + 3x − 28, find factor pairs of −28: (1, −28), (−1, 28), (2, −14), (−2, 14), (4, −7), and (−4, 7). The pair (−4, 7) sums to 3, so the factorization is (x − 4)(x + 7).

Similarly, for x² − 11x + 30, factor pairs of 30 are (1, 30), (2, 15), (3, 10), and (5, 6). Since the sum must be −11, use (−5, −6), which multiply to 30 and add to −11. The factorization is (x − 5)(x − 6).

Always verify your factorization by expanding the binomials to ensure they produce the original trinomial. This method applies only when the coefficient of x² is 1. For trinomials with different leading coefficients, other factoring techniques are required.

Mastering this factoring technique enhances your ability to solve quadratic equations, simplify expressions, and understand polynomial functions, forming a foundation for more advanced algebraic concepts.

Problem

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

$(y + 3) (y - 4)$

$(y - 3) (y - 4)$

$(y + 7) (y - 1)$

$(y - 1) (y - 12)$

Problem

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

$(z + 5) (z - 6)$

$(z - 3) (z - 10)$

$(z + 5) (z + 6)$

$(z - 5) (z - 6)$

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 this, first identify pairs of factors of the constant term, which is 6y². Possible factor pairs include 1y and 6y, or 2y and 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 this factorization, apply the FOIL method:

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 matches the original trinomial, confirming the factorization is correct. This process highlights how factoring techniques extend naturally to polynomials with multiple variables by treating variable terms as part of the factors and carefully considering their coefficients.

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 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 to multiply two binomials. When you multiply $(x + p) (x + q)$ , FOIL gives $x^{2} + (p+q)x + pq$ . Factoring reverses this: given $x^{2} + bx + c$ , find two numbers $p$ and $q$ such that $p + q = b$ and $pq = c$ . This connection between multiplication and factoring is key to solving trinomials of this form.

When the constant term $c$ is negative in a trinomial $x^{2} + bx + c$ , you look for two numbers that multiply to a negative number and add to $b$ . One number will be positive and the other negative. To find these, list factor pairs of the absolute value of $c$ and assign signs so their sum equals $b$ . For example, for $x^{2} + 3x - 28$ , factor pairs of 28 are (1, 28), (2, 14), (4, 7). Assign signs to find which pair sums to 3: $-4 + 7 = 3$ . So, the factorization is $(x - 4)(x + 7)$ .

The method described for factoring trinomials of the form $x^{2} + bx + c$ applies only when the coefficient of $x^{2}$ is 1. If the coefficient is not 1, such as in $ax^2 + bx + c$ where $a 2 1$ , you need different techniques like factoring by grouping or using the quadratic formula. The process involves more steps because the product and sum conditions change. So, this simple factoring method is limited to trinomials with leading coefficient 1.

To systematically find the two numbers that multiply to $c$ and add to $b$ in a trinomial $x^{2} + bx + c$ , use a T-chart to list all factor pairs of $c$ . For each pair, calculate their sum. The pair whose sum equals $b$ is the one you need. For example, for $x^2 + 3x - 28$ , list factor pairs of -28: (1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), (-4, 7). Check sums: only $-4 + 7 = 3$ matches $b$ . This method helps organize your work and avoid mistakes.