Question

do you understand?

1. reasoning explain how to use the foil method to find the product of two binomials.
2. compare and contrast simplify (3x + 8)(x + 1) using a table, the distributive property, and the foil method. which method is most efficient? explain.
3. writing how is the degree of the product of two polynomials p(x) and q(x) related to the degrees of p(x) and q(x)?

Explanation:

Question 15

FOIL is an acronym for First, Outer, Inner, Last, a step-by-step way to multiply two binomials:

1. First: Multiply the first terms of each binomial.
2. Outer: Multiply the outer (leftmost and rightmost) terms of the binomials.
3. Inner: Multiply the inner (rightmost of first, leftmost of second) terms.
4. Last: Multiply the last terms of each binomial.
5. Combine like terms (if any) to simplify the final expression.

For example, for $(a+b)(c+d)$:

• First: $a \times c = ac$
• Outer: $a \times d = ad$
• Inner: $b \times c = bc$
• Last: $b \times d = bd$
• Simplified: $ac + ad + bc + bd$

Step1: Table Method

Create a table to multiply each term:

	$3x$	$8$
$1$	$3x$	$8$

Sum all terms: $3x^2 + 8x + 3x + 8 = 3x^2 + 11x + 8$

Step2: Distributive Property

Apply $a(b+c)=ab+ac$ twice:
First, treat $(3x+8)$ as $a$:
$$(3x+8)(x+1) = (3x+8)x + (3x+8)1$$
Expand each term:
$$= 3x^2 + 8x + 3x + 8$$
Combine like terms:
$$= 3x^2 + 11x + 8$$

Step3: FOIL Method

Apply First, Outer, Inner, Last:

• First: $3x \times x = 3x^2$
• Outer: $3x \times 1 = 3x$
• Inner: $8 \times x = 8x$
• Last: $8 \times 1 = 8$

Combine terms:
$$3x^2 + 3x + 8x + 8 = 3x^2 + 11x + 8$$

Step4: Efficiency Comparison

The FOIL method is most efficient for binomial multiplication because it directly targets the four required multiplications in a single, memorizable sequence, with no extra setup (like a table) or repeated distributive steps.

When multiplying two polynomials, the highest-degree term of the product comes from multiplying the highest-degree terms of each original polynomial. If $p(x)$ has degree $m$ and $q(x)$ has degree $n$, their product's highest-degree term is $x^m \times x^n = x^{m+n}$.

Answer:

Use the FOIL acronym steps: Multiply First terms, Outer terms, Inner terms, Last terms, then combine like terms.

---

Question 16