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:

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.