Questions and Answers #84 Polynomial Factorization

Questions and Answers Sheet 84 Polynomial Factorization Question #1 How do you multiply 4𝑥(2𝑥 2 + 𝑥)? Answer: Explanation: You would do this by distributing 4x into the quantities in the parenthesis. For example, 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 The a was distributed into the parenthesis and multiplied by each unit inside of it. The same concept applies here, we just have expressions instead of constant numbers. First, distribute the 4x, 4𝑥(2𝑥 2 + 𝑥) = (4𝑥 ∗ 2𝑥 2 ) + (4𝑥 ∗ 𝑥) You don't need to add parenthesis, I just added them to clarify how the 4x was distributed. Now you can simplify. We can use exponent rules here. 𝑥 𝑎 ∗ 𝑥 𝑏 = 𝑥 𝑎+𝑏 Using this, we can simplify further. Note that the constant coefficients just get multiplied normally. 4𝑥(2𝑥 2 + 𝑥) = (4𝑥 1 ∗ 2𝑥 2 ) + (4𝑥 1 ∗ 𝑥 1 ) = 8𝑥 1+2 + 4𝑥 1+1 = 8𝑥 3 + 4𝑥 2 Question #2 What is the standard form of a polynomial 7𝑥(3 − 𝑥 + 6𝑥 3 )? Answer: Multiply each term in parenthesis by 7x and then arrange the resulting terms in descending exponent size: (7𝑥 × 3) − (7𝑥 × 𝑥) + (7𝑥 × 6𝑥 3 ) → 21𝑥 − 7𝑥 2 + 42𝑥 4 42𝑥 4 − 7𝑥 2 + 21𝑥 Question #3 What is the standard form of a polynomial 5𝑦(𝑦 5 + 8𝑦 3 )? Answer: Explanation: The first step is to distribute the bracket. ⇒ 5𝑦(𝑦 5 + 8𝑦 3 ) = 5𝑦 6 + 40𝑦 4 in standard form expressing a polynomial in standard form means writing the term with the highest power of the variable, followed by descending powers of the variable until the last term, usually a constant. Here there are only 2 terms. The one with the highest power of the variable being 5𝑦 6 Answer: 5𝑦 6 + 40𝑦 4 Question #4 How do you simplify (𝑎 + 4𝑏 − 3𝑐)3 ? Answer: Explanation: Take it step-by-step, using the fact that when you multiply one polynomial times another, each of the terms in the first one needs to get multiplied one time by each of the terms in the second one (this follows from the distributive property). So, first we compute (𝑎 + 4𝑏 − 3𝑐)2 : (𝑎 + 4𝑏 − 3𝑐)2 = (𝑎 + 4𝑏 − 3𝑐) ∗ (𝑎 + 4𝑏 − 3𝑐) = 𝑎2 + 4𝑎𝑏 − 3𝑎𝑐 + 4𝑎𝑏 + 16𝑏 2 − 12𝑏𝑐 − 3𝑎𝑐 − 12𝑏𝑐 + 9𝑐 2 Combine like terms to get: (𝑎 + 4𝑏 − 3𝑐)2 = 𝑎2 + 16𝑏 2 + 9𝑐 2 + 8𝑎𝑏 − 6𝑎𝑐 − 24𝑏𝑐 Now (𝑎 + 4𝑏 − 3𝑐)3 = (𝑎 + 4𝑏 − 3𝑐) ∗ (𝑎2 + 16𝑏 2 + 9𝑐 2 + 8𝑎𝑏 − 6𝑎𝑐 − 24𝑏𝑐) = 𝑎3 + 16𝑎𝑏 2 + 9𝑎𝑐 2 − 6𝑎2 𝑐 − 24𝑎𝑏𝑐 + 4𝑎2 𝑏 + 64𝑏 3 + 36𝑏𝑐 2 + 32𝑎𝑏 2 −24𝑎𝑏𝑐 − 96𝑏 2 𝑐 − 3𝑎2 𝑐 − 48𝑏 2 𝑐 − 27𝑐 3 − 24𝑎𝑏𝑐 + 18𝑎𝑐 2 + 72𝑏𝑐 2 = 𝑎3 + 12𝑎2 𝑏 + 48𝑎𝑏 2 + 64𝑏 3 − 9𝑎2 𝑐 − 72𝑎𝑏𝑐 − 144𝑏 2 𝑐 + 27𝑎𝑐 2 + 108𝑏𝑐 2 − 27𝑐 3 Question #5 What is the standard form of a polynomial 6𝑥(𝑥 2 + 2𝑥 + 1)? Answer: Explanation: 1. Use the distributive property, 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐, to multiply 6x by each term within the brackets. 6𝑥(𝑥 2 + 2𝑥 + 1) = 6𝑥(𝑥 2 ) + 6𝑥(2𝑥) + 6𝑥(1) 2. Simplify. = 6𝑥 3 + 12𝑥 2 + 6𝑥 Question #6 How do you write 10𝑥(9𝑥 2 + 6𝑥 + 7) in standard form? Answer: Explanation: Multiply everything inside the rackets by 10x ⇒ 𝑦 = 90𝑥 3 + 60𝑥 2 + 70𝑥 If it is required that the constant term is included then you have: ⇒ 𝑦 = 90𝑥 3 + 60𝑥 2 + 70𝑥 + 0 (Ans) Question #7 How do you write 2𝑥 4 (4𝑥 2 + 3𝑥 + 1) in standard form? Answer: Explanation: Multiply each term within the parenthesis by the term outside the parenthesis: 2𝑥 4 (4𝑥 2 + 3𝑥 + 1) ⇒ (2𝑥 4 × 4𝑥 2 ) + (2𝑥 4 × 3𝑥) + (2𝑥 4 × 1) ⇒ 8𝑥 6 + 6𝑥 5 + 2𝑥 4 Question #8 How do you write (𝑥 2 + 4𝑥 − 3)(2𝑥 2 + 𝑥 + 6) in standard form? Answer: Explanation: This is a long one, but it's not difficult, as long as we watch our signs (𝑥 2 + 4𝑥 − 4)(2𝑥 2 + 𝑥 + 6) 2𝑥 4 + 𝑥 3 + 6𝑥 2 + 8𝑥 3 + 4𝑥 2 + 24𝑥 − 6𝑥 2 − 3𝑥 − 18 combine like terms: 2𝑥 4 + (1𝑥 3 + 8𝑥 3 ) + (6𝑥 2 + 4𝑥 2 − 6𝑥 2 ) + (24𝑥 − 3𝑥) − 18 2𝑥 4 + 9𝑥 3 + 4𝑥 2 + 21𝑥 − 18 Question #9 How do you multiply (3𝑥 2 + 2𝑥 + 4)(2𝑥 + 1)? Answer: Explanation: You have to take each number in the first bracket and multiply it by the second one. First lets do 3𝑥 2 (3𝑥 2 ) ∗ (2𝑥) and (3𝑥 2 ) ∗ 1 This will give 6𝑥 3 and 3𝑥 2 Then 2x 2𝑥 ∗ 2𝑥 and 2𝑥 ∗ 1 This will give 4𝑥 2 and 2x Then 4 4 ∗ 2𝑥 and 4 ∗ 1 This will give 8x and 4 Question #10 How do you multiply (𝑥 2 + 3𝑥)(𝑥 + 1)? Answer: Explanation: multiplying out: 𝑥 3 + 3𝑥 2 + 𝑥 2 + 3𝑥 = 𝑥 3 + 4𝑥 2 + 3𝑥 𝑥 3 + 4𝑥 2 + 3𝑥 (Ans)