If p(x) = 5x^2 - 4x + 3, what is p(x) + p(-x)?
Answer and explanation
Correct answer: 10x^2 + 6
The governing concept is substitution of -x and cancellation of odd-power terms. Starting with p(x)=5x^2-4x+3, we get p(-x)=5(-x)^2-4(-x)+3=5x^2+4x+3. The square term is unchanged because (-x)^2=x^2, while the linear term changes sign. Adding gives p(x)+p(-x)=(5x^2-4x+3)+(5x^2+4x+3)=10x^2+6. Thus option A is correct. The terms -4x and +4x cancel, while the two quadratic terms and the two constants add. Option B incorrectly keeps the linear term, C omits one quadratic contribution, and D incorrectly removes the quadratic terms. This also illustrates that the sum of a polynomial and its reflection in x is even.
Frequently asked questions
What is the correct answer to this question?
10x^2 + 6
Why is this the correct answer?
The governing concept is substitution of -x and cancellation of odd-power terms. Starting with p(x)=5x^2-4x+3, we get p(-x)=5(-x)^2-4(-x)+3=5x^2+4x+3. The square term is unchanged because (-x)^2=x^2, while the linear term changes sign. Adding gives p(x)+p(-x)=(5x^2-4x+3)+(5x^2+4x+3)=10x^2+6. Thus option A is correct. The terms -4x and +4x cancel, while the two quadratic terms and the two constants add. Option B incorrectly keeps the linear term, C omits one quadratic contribution, and D incorrectly removes the quadratic terms. This also illustrates that the sum of a polynomial and its reflection in x is even.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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