(x + 4) is a factor of which one of the following expressions?
x 2– 7x – 44
Understanding Factors of Polynomials
A factor of an expression (like a polynomial) is another expression that divides into it evenly, leaving no remainder. In other words, if (x + 4) is a factor of an expression, then the value of the expression is 0 when x = -4. This is based on the Factor Theorem.
The Factor Theorem states that for a polynomial P(x), (x - a) is a factor if and only if P(a) = 0. In this question, we are checking if (x + 4) is a factor. This means we need to check if the expression equals 0 when x = -4 (since x + 4 = x - (-4), so a = -4).
Evaluating Each Expression Using the Factor Theorem
We will substitute x = -4 into each of the given expressions and see which one results in 0.
Evaluating Expression 1: \({x}^{2} - 7x + 44\)
Substitute x = -4:
\[{(-4)}^{2} - 7(-4) + 44\] \[= 16 - (-28) + 44\] \[= 16 + 28 + 44\] \[= 44 + 44\] \[= 88\]
Since the result is 88 and not 0, (x + 4) is not a factor of \({x}^{2} - 7x + 44\).
Evaluating Expression 2: \({x}^{2} + 7x - 44\)
Substitute x = -4:
\[{(-4)}^{2} + 7(-4) - 44\] \[= 16 + (-28) - 44\] \[= 16 - 28 - 44\] \[= -12 - 44\] \[= -56\]
Since the result is -56 and not 0, (x + 4) is not a factor of \({x}^{2} + 7x - 44\).
Evaluating Expression 3: \({x}^{2} - 7x - 44\)
Substitute x = -4:
\[{(-4)}^{2} - 7(-4) - 44\] \[= 16 - (-28) - 44\] \[= 16 + 28 - 44\] \[= 44 - 44\] \[= 0\]
Since the result is 0, (x + 4) is a factor of \({x}^{2} - 7x - 44\).
Evaluating Expression 4: \({x}^{2} + 7x + 44\)
Substitute x = -4:
\[{(-4)}^{2} + 7(-4) + 44\] \[= 16 + (-28) + 44\] \[= 16 - 28 + 44\] \[= -12 + 44\] \[= 32\]
Since the result is 32 and not 0, (x + 4) is not a factor of \({x}^{2} + 7x + 44\).
Conclusion
Based on the evaluations, only the expression \({x}^{2} - 7x - 44\) results in 0 when x = -4. Therefore, (x + 4) is a factor of this expression.
Summary of Results
- \({x}^{2} - 7x + 44\) evaluated at x = -4 is 88.
- \({x}^{2} + 7x - 44\) evaluated at x = -4 is -56.
- \({x}^{2} - 7x - 44\) evaluated at x = -4 is 0.
- \({x}^{2} + 7x + 44\) evaluated at x = -4 is 32.
Revision Table: Polynomial Factors
Here is a quick table summarizing the results:
| Expression | Value at x = -4 | Is (x + 4) a Factor? |
|---|---|---|
| \({x}^{2} - 7x + 44\) | 88 | No |
| \({x}^{2} + 7x - 44\) | -56 | No |
| \({x}^{2} - 7x - 44\) | 0 | Yes |
| \({x}^{2} + 7x + 44\) | 32 | No |
Additional Information: The Factor Theorem and Roots
The Factor Theorem is a direct consequence of the Remainder Theorem. The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a).
- If P(a) = 0, the remainder is 0, which means (x - a) divides P(x) evenly. Thus, (x - a) is a factor.
- If (x - a) is a factor of P(x), it means P(x) = (x - a) * Q(x) for some polynomial Q(x). If we substitute x = a, we get P(a) = (a - a) * Q(a) = 0 * Q(a) = 0.
The value 'a' for which P(a) = 0 is also called a root or a zero of the polynomial P(x). So, (x + 4) being a factor means that x = -4 is a root of the polynomial.
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