The function p (x) is an odd degree polynomial with a negative leading coefficient. If q (x) = x3 + 5x2 - 9x - 45, which statement is true?

As x approaches negative infinity, p (x) approaches positive infinity and q (x) approaches negative infinity.

As x approaches negative infinity, p (x) and q (x) approach positive infinity.

As x approaches negative infinity, p (x) and q (x) approach negative infinity.

As x approaches negative infinity, p (x) approaches negative infinity and q (x) approaches positive infinity.

Question

Mathematics

Kamren

20 June, 06:19

The function p (x) is an odd degree polynomial with a negative leading coefficient. If q (x) = x3 + 5x2 - 9x - 45, which statement is true?

As x approaches negative infinity, p (x) approaches positive infinity and q (x) approaches negative infinity.

As x approaches negative infinity, p (x) and q (x) approach positive infinity.

As x approaches negative infinity, p (x) and q (x) approach negative infinity.

As x approaches negative infinity, p (x) approaches negative infinity and q (x) approaches positive infinity.

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Gromit · Answer 1

As x approaches negative infinity, p (x) approaches positive infinity and q (x) approaches negative infinity. The function q (x) given is an odd degree polynomial with a positive leading coefficient. Because odd degree polynomials preserve the sign of the value of x, and because the leading coefficients of both functions p (x) and q (x) are of opposite signs, they will diverge to opposite infinities as x approaches infinity. So let's look at the options and see what makes sense. As x approaches negative infinity, p (x) approaches positive infinity and q (x) approaches negative infinity. * OK. This has p (x) and q (x) going to different infinities. Negative infinity to an odd power will result in a negative infinity. And since q (x) has a positive leading coefficient, that means that q (x) will approach negative infinity. Everything here matches, so this is the correct choice. As x approaches negative infinity, p (x) and q (x) approach positive infinity. * This is claiming that p (x) and q (x) are approaching the same infinity. So we immediately know this is the wrong choice, given what I said earlier about those functions. As x approaches negative infinity, p (x) and q (x) approach negative infinity. * This is claiming that p (x) and q (x) are approaching the same infinity. So we immediately know this is the wrong choice, given what I said earlier about those functions. As x approaches negative infinity, p (x) approaches negative infinity and q (x) approaches positive infinity. * OK. This has p (x) and q (x) going to different infinities. Negative infinity to an odd power will result in a negative infinity. And since q (x) has a positive leading coefficient, that means that q (x) will approach negative infinity. But this option is claiming that q (x) is going to positive infinity. That's wrong and therefore this choice is wrong.