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Algebra II – Polynomial Function

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    Study Guide Algebra II – Polynomial Function 1. Polynomial Function What Is a Polynomial Function? A polynomial function is a type of mathematical expression made up of variables and numbers combined using addition, subtraction, and multiplication . The general form of a polynomial function looks like this: Understanding the Parts • (x) is the variable. • (a 0 , a 1 , a 2 , … , a n ) are numbers called coefficients . • (n) is a whole number (0, 1, 2, 3, …). • The highest power of (x) determines the degree of the polynomial. Polynomial functions are evaluated by substituting a number for the variable (x) and then simplifying the expression. For example, if we replace (x) with 4 in (P(x)), we write it as: This means “find the value of the polynomial when (x = 4)” . Example 1: Evaluating a Polynomial Suppose we have the polynomial:

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    Study Guide We want to find (P( - 4)) . Step 1: Substitute (x = - 4) Step 2: Calculate the powers Step 3: Multiply Step 4: Add and subtract This means when (x = - 4), the value of the polynomial is - 241 . Example 2: Finding (f(x + h)) Now consider the function: We want to find (f(x + h)) . This means we replace every (x) with (x + h) . Step 1: Substitute (x + h)

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    Study Guide Step 2: Expand the square So, Step 3: Multiply Final Expression This is the expanded form of (f(x + h)) . 2. Quiz: Polynomial Function 1. Question If (P(y) = 2y 4 - 3y 3 + 4y 2 - 7), find (P(3)). Answer Choices • 110 • 81 • 29 Correct Answer : 110

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    Study Guide Why This Is Correct To find (P(3)), substitute (y = 3) into the polynomial. Now calculate each term: Therefore, the correct answer is 110 . 2. Question If (Q(z) = 3z 3 + 4z 2 - 7z + 8), find (Q( - 5)). Answer Choices • 232 • 518 • - 232 Correct Answer : - 232 Why This Is Correct Substitute (z = - 5) into the polynomial. Calculate each part:

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    Study Guide So the correct answer is - 232 . 3. Question If (f(x) = x 3 + 6x + 9), find (f(x - 2)). Answer Choices • (x 3 + 6x 2 + 8x + 9) • (x 3 - 6x 2 + 18x - 11) • (x 3 - 6x 2 - 8x + 9) Correct Answer : (x 3 - 6x 2 + 18x - 11) Why This Is Correct Substitute (x - 2) into the function. First expand the cube: Now substitute: Combine like terms:

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    Study Guide Thus the correct answer is (x 3 - 6x 2 + 18x - 11) . 4. Question If (f(x) = x 2 - 4x - 3), find (f(x + 3)). Answer Choices • (x 2 - 2x - 6) • (x 2 + 2x - 6) • (x 2 + 2x + 6) Correct Answer : (x 2 + 2x - 6) Why This Is Correct Substitute (x + 3) into the function. Expand: Combine like terms: So the correct answer is (x 2 + 2x - 6) .

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    Study Guide 5. Question If (g(x) = 2x 3 + 4x - 5), find (g(x - m)). Answer Choices • (2x 3 - m 2 x 2 + 4x - 4m - 2m 3 - 5) • (2x 3 + 12m 2 x 2 + 4x - 4m - 2m 3 - 5) • (2x 3 - 6mx 2 + 6m 2 x + 4x - 4m - 2m 3 - 5) Correct Answer : (2x 3 - 6mx 2 + 6m 2 x + 4x - 4m - 2m 3 - 5) Why This Is Correct Substitute (x - m) into the function. First expand the cube: Multiply by 2: Now include the remaining terms: Final expression: Thus the correct answer is (2x 3 - 6mx 2 + 6m 2 x + 4x - 4m - 2m 3 - 5) .

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    Study Guide 3. Remainder Theorem What Is the Remainder Theorem? The Remainder Theorem helps us quickly find the remainder when a polynomial is divided by a linear expression. It states: If a polynomial (P(x)) is divided by ((x - r)), then the remainder is equal to (P(r)) . In simple terms Instead of performing long division, you can: 1. Replace (x) with (r) in the polynomial. 2. Calculate the value. 3. That value is the remainder . So, This makes calculations much faster and easier . Example 1: Finding (P( - 3)) Given: We want to find:

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    Study Guide There are two possible methods to solve this. Method 1: Direct Substitution Simply replace (x) with − 3 . Step 1: Evaluate the powers Step 2: Substitute the values Step 3: Multiply Step 4: Simplify Final Answer: Method 2: Using Synthetic Division

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    Study Guide Another way to find the remainder is by synthetic division . Since we want (P( - 3)), we divide the polynomial by: which simplifies to: Using synthetic division with − 3 , we calculate the remainder. After completing the synthetic division steps, the remainder is −1610 . This confirms our result from Method 1 . Therefore, Example 2: Finding the Remainder Find the remainder when: is divided by: Method 1: Apply the Remainder Theorem According to the theorem, Step 1: Substitute (x = 4)

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