Identifying which Algebraic Expressions are even

Transcript

Identifying which Algebraic Expressions are even Identify which of the following algebraic expressions give an even number for any integer values for the letter-nWe follow the same cheatsheet as before. Let’s quickly revise it Parity rules (Cheat Sheet) Addition/Subtraction: Odd ± Odd = Even (e.g., 3 + 5 = 8) Even ± Even = Even (e.g., 2 + 4 = 6) Odd ± Even = Odd (e.g., 3 + 2 = 5) Multiplication: If you multiply any number by an Even number, the result is always Even. Odd × Odd is the only way to get Odd. Powers: An Odd number to any power is Odd. An Even number to any power is Even. Now, let’s do the questions, one-by-one Note that: Any term multiplied by an even number (like 2𝑎,4𝑚,6𝑘 ) is guaranteed to be Even. 2𝑎+2𝑏 Analysis: 2𝑎 is Even. 2𝑏 is Even. Logic: Even + Even = Even. Verdict: ALWAYS EVEN 3𝑔+5ℎ Analysis: If 𝑔 is 1 and ℎ is 2 , you get 3(1)+5(2)=3+10=13 (Odd). Verdict: Not always even. 4𝑚+2𝑛 Analysis: 4𝑚 is a multiple of 4 (Even). 2𝑛 is a multiple of 2 (Even). Logic: Even + Even = Even. Verdict: ALWAYS EVEN 2𝑢−4𝑣 Analysis: 2𝑢 is Even. 4𝑣 is Even. Logic: Even - Even = Even. Verdict: ALWAYS EVEN 13𝑘−5𝑘 Analysis: Simplify the algebra first! 13𝑘−5𝑘=8𝑘. Logic: Any number multiplied by 8 is Even. Verdict: ALWAYS EVEN 6𝑚−3𝑛 Analysis: 6𝑚 is Even. But 3𝑛 could be Odd (if 𝑛=1,3𝑛=3 ). Logic: Even - Odd = Odd. It fails for some numbers. Verdict: Not always even. 𝑥^2+2 Analysis: If 𝑥 is 1 (odd), 1^2+2=3 (Odd). (As explained in the textbook image). Verdict: Not always even. 𝑏^2+1 Analysis: If 𝑏 is 2 (even), 2^2+1=5 (Odd). Verdict: Not always even. 4𝑘 × 3𝑗 Analysis: This is multiplication. One of the factors is 𝟒(4𝑘). Logic: Since there is an even factor, the entire product is Even. Verdict: ALWAYS EVEN Answer Summary: The expressions that are always even are 2𝑎+2𝑏,4𝑚+2𝑛,2𝑢−4𝑣, 13𝑘−5𝑘, and 4𝑘×3𝑗.