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Consider the polynomial function 𝑓 of 𝑥 is equal to negative eight 𝑥 to the five plus three 𝑥 to the four minus 12𝑥 to the six plus five 𝑥 squared minus 12.
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What is its degree?
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And what is its leading coefficient?
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The degree of a one-variable polynomial is the highest power or exponent of that variable.
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The powers or exponents in this polynomial function are five, four, six, and two.
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We couldn’t factor at the final term, the constant term, as negative 12 multiplied by 𝑥 to the zero, as 𝑥 to the power of zero is equal to one.
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The highest power of 𝑥 — it appears in this polynomial function — is six.
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And so this is the degree of the polynomial.
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Now let’s consider the leading coefficient.
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The leading coefficient is the number in front of the variable with the highest power.
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We’ve already established that the term with the highest power is the term with the 𝑥 to the six.
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And the coefficient here is negative 12.
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Note that the sign of this number is important.
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The coefficient isn’t 12.
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It’s negative 12.
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It is perhaps more usual to see polynomial functions written in order of decreasing powers of 𝑥.
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If this were the case, then the leading coefficient is in fact the first coefficient in the polynomial function.
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However, as we’ve seen in this question, this isn’t always the case.
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So we need to look carefully through the polynomial in order to find the term with the highest power.
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The degree of this polynomial function is six.
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And the leading coefficient is negative 12.