Question

If P(x) = 12x^3 + 11x^2 - 9x + 10 and Q(x) = 4x^2 – 3x + 2, find P(x) + Q(x).

Answer 1

To find P(x) + Q(x), add the corresponding coefficients of like terms from P(x) = 12x^3 + 11x^2 - 9x + 10 and Q(x) = 4x^2 – 3x + 2. The result is a new polynomial, R(x) = 12x^3 + 15x^2 – 12x + 12. This process involves simple addition of coefficients.

Step-by-step explanation:

To find P(x) + Q(x), we simply add the corresponding coefficients of the like terms from each polynomial. Let's go through the process step by step:

1. Write down the two polynomials P(x) = 12x3 + 11x2 - 9x + 10 and Q(x) = 4x2 – 3x + 2.
2. Arrange the terms so that like terms are lined up with each other. In this case, P(x) has a cubic term, which Q(x) does not have, so we will write it first. For the rest of the terms, we align the squared terms, the x terms, and the constant terms.
3. Add the coefficients of the like terms. The cubic term is only present in P(x), so it remains unchanged. For the squared terms, we add the coefficients 11 and 4, for the x terms, we add -9 and -3, and for the constants, we add 10 and 2.
4. The result gives us the new polynomial, which is the sum of P(x) and Q(x): R(x) = 12x3 + (11+4)x2 + (-9-3)x + (10+2).
5. Simplify the coefficients: R(x) = 12x3 + 15x2 - 12x + 12.

Therefore, the sum P(x) + Q(x) is 12x3 + 15x2 - 12x + 12.