Problem 72 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(b^{2}+14 b+48\)

Step-by-Step Solution

Answer

(x+6)(x+8)

Step 1: Identify the Coefficients

Recognize that the coefficients are: - Coefficient of the middle term: \(b = 14\) - Constant term: \(c = 48\)

Step 2: Find Two Numbers that Multiply to c and Add to b

Identify two numbers that multiply to \(48\) and add to \(14\). a. List the factor pairs of 48: \((1, 48), (2, 24), (3, 16), (4, 12), (6, 8)\)b. Check which pair adds to 14. Only \((6, 8)\) sums to \(14\).

Step 3: Write the Factored Form

The factored form of the trinomial is found by writing the expression as the product of two binomials: \((x+6)(x+8)\)

Step 4: Verify the Solution

Expand parentheses to verify:\( (x+6)(x+8)\rightarrow x^{2}+8x +6x+48 \rightarrow x^{2}+14x+ 48 \) to ensure it matches the original trinomial.

Key Concepts

coefficient identification

Understanding how to identify the coefficients in a trinomial is crucial for factoring it correctly. The general form of a trinomial is \text: \(ax^2 + bx + c\). In this form:

• \(a\) is the coefficient of \(x^2\) (which is 1 in our case)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

In the trinomial \(x^2 + 14x + 48\), the coefficient of \(x\) (\(b\)) is 14, and the constant term (\(c\)) is 48. Recognizing these coefficients is the first step in the factoring process.

factor pairs

The next step involves finding factor pairs. Factor pairs are two numbers that multiply to give you one value and add to give you another value. For our trinomial \(x^2 + 14x + 48\), we need pairs that multiply to 48 (\(c\)) and add up to 14 (\(b\)). First, list the factor pairs of 48:

• (1, 48)
• (2, 24)
• (3, 16)
• (4, 12)
• (6, 8)

Upon examination, only the pair (6, 8) adds up to 14. Therefore, 6 and 8 are the numbers we need to use in the next steps.

binomial products

Now that we have our two numbers, we can express the trinomial as a product of binomials. If our two numbers that we found are 6 and 8, we write the factored form of the trinomial as: \text: \((x + 6)(x + 8)\). These binomials \(\text{(x + 6)}\) and \(\text{(x + 8)}\) represent the solution to the trinomial \(\text{x² + 14x + 48}\).

trinomial verification

After writing the trinomial in its factored form, it is important to verify the solution to ensure correctness. Verification is done by expanding the binomials and checking if the result matches the original trinomial:- Expand \((x + 6)(x + 8)\):

• First, \(x \times x\) gives \(x^2\).


• Next, \(x \times 8\) gives \(8x\).


• Then, \(6 \times x\) equals \(6x\).


• Finally, \(6 \times 8\) results in \(48\).

Adding these together, we get \(x^2 + 8x + 6x + 48\). Combining like terms results in \(x^2 + 14x + 48\), which matches the original trinomial.Hence, the factored form \((x + 6)(x + 8)\) is indeed correct.