SAT-Math-28: Multiplying Polynomials (FOIL and beyond)

1. The Core Mechanic: Total Distribution

Multiplying polynomials is essentially the distributive property on repeat. Every single term in the first polynomial must be multiplied by every single term in the second polynomial.

Binomial Multiplication: The FOIL Method

When multiplying two binomials (two-term expressions), use the classic acronym FOIL to ensure you don't miss a combination:

• First: Multiply the first terms of each binomial.

• Outer: Multiply the outermost terms.

• Inner: Multiply the innermost terms.

• Last: Multiply the last terms of each binomial.

$$(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}$$

🇺🇿 Uzbek Explanation:

Ko'phadlarni ko'paytirish (FOIL usuli): Ikkita qavsni bir-biriga ko'paytirishda birinchi qavsning ichidagi har bir hadni, ikkinchi qavsning ichidagi har bir hadga ketma-ket ko'paytirib chiqish kerak. Hech bir had e'tibordan chetda qolmasligi lozim! Keyin esa hosil bo'lgan o'xshash hadlarni ixchamlaymiz.

2. High-Yield SAT Practice Question

Question: Which of the following is equivalent to the expression $(2x + 3)(x - 5)$?

A) $2x^2 - 15$

B) $2x^2 - 7x - 15$

C) $2x^2 + 7x - 15$

D) $2x^2 - 10x - 15$

Step-by-Step Explanation:

1. Apply the FOIL method step-by-step:

   • First: $2x \cdot x = 2x^2$

   • Outer: $2x \cdot (-5) = -10x$

   • Inner: $3 \cdot x = 3x$

   • Last: $3 \cdot (-5) = -15$

2. Write out the expanded terms side-by-side:

   $$2x^2 - 10x + 3x - 15$$

3. Combine the like terms in the middle: Combine the $x$ terms ($-10x + 3x = -7x$).

   $$2x^2 - 7x - 15$$

Correct Answer: B

Summary

• FOIL stands for First, Outer, Inner, Last—your structural safety net for multiplying binomials.

• Keep absolute track of negative signs during multiplication (e.g., positive times negative yields negative).

• Always combine like terms at the final step to present your polynomial in standard form.

Wonderful progress, my bro! Are you ready to dive into the reverse process with SAT-Math-29: Factoring: Greatest Common Factor (GCF) and Grouping?