Question

ap precalculus
due tue, oct 14, 2025
degree / leading coefficient / end behavior
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1. $y = x(2x - 1)^{10}$

Explanation:

Step1: Expand using binomial theorem concept

The binomial expansion of \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). Here \((2x-1)^{10}=\sum_{k = 0}^{10}\binom{10}{k}(2x)^{10 - k}(-1)^{k}\). But we can also find the degree and leading - coefficient in another way. When we multiply \(x\) by \((2x-1)^{10}\), the highest - degree term of \((2x - 1)^{10}\) is \((2x)^{10}\) (using the first term of the binomial expansion when \(k = 0\)).

Step2: Find the degree of the polynomial

When we multiply \(x\) by \((2x)^{10}\), we use the rule of exponents \(a^m\times a^n=a^{m + n}\). So the degree of \(y=x(2x - 1)^{10}\) is \(1+10=11\) since the degree of \(x\) is \(1\) and the degree of \((2x)^{10}\) is \(10\).

Step3: Find the leading coefficient

The leading - term of \(y=x(2x - 1)^{10}\) comes from multiplying \(x\) by the leading - term of \((2x-1)^{10}\). The leading - term of \((2x-1)^{10}\) is \((2x)^{10}=2^{10}x^{10}\), and when we multiply it by \(x\), we get \(x\times2^{10}x^{10}=2^{10}x^{11}\). Since \(2^{10}=1024\), the leading coefficient is \(1024\).

Step4: Determine the end - behavior

For a polynomial \(y = a_nx^n+\cdots+a_0\) with \(n = 11\) (an odd number) and \(a_n=1024>0\), as \(x\to-\infty\), \(y\to-\infty\) (because \(y=a_nx^n\) dominates for large \(|x|\) and when \(x\) is negative and \(n\) is odd, \(x^n\) is negative and \(a_n>0\)), and as \(x\to+\infty\), \(y\to+\infty\) (because when \(x\) is positive and \(n\) is odd, \(x^n\) is positive and \(a_n>0\)).

Answer:

Degree: 11; Leading Coefficient: 1024; End - behavior: As \(x\to-\infty\), \(y\to-\infty\); as \(x\to+\infty\), \(y\to+\infty\)