Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10 or (x 10) End Behavior Solving Factoring Graphing End Behavior real root x=0 or (x+0) local minimum Synthetic Division 2 Long Division Rational Zeros
Properties of Exponents P1
Adding and Subtracting Polynomials Adding & Subtracting Polynomials is COMBINING LIKE TERMS. To be considered like terms, the terms must have the same variable, and the variables must have the same exponents. We add or subtract the coefficients, leaving the variables unchanged. All answers must be written in standard form, the largest exponent first, the rest in descending order. Examples: Horizontal Format: Vertical Format:
Multiplication of Polynomials Distributive Property Multiply both terms in the second parenthesis by the first term in the first parenthesis. Next, multiply each term in the second parenthesis by the second term in the first parenthesis. Combine like terms. Vertical Method Write equations in standard form. Align like terms in columns. Multiply each term in the top equation by each term in the bottom equation. Combine like terms.
Factoring Polynomials P2 Factor by Grouping Algorithm: 1. Split the problem into two parts. 2. Factor the GCF out of each part. 3. Factor the common parentheses out. 4. Write as the product of two binomials.
Long Division Algorithm: 1. Write the function in standard form. (Exponents in descending order, allow zeros as place holders) 2. Make the leading term of the divisor exactly match the leading term of the dividend using multiplication. 3. Distribute the factor through the divisor and subtract. (Subtract every term) 4. Repeat steps 2 and 3 if necessary. 5. Write any remainders in fraction form. Example: Check:
Synthetic Substitution All equations must be written with decreasing exponents, biggest exponent first, constant last. Rearrange if necessary = Make sure every exponent has a spot, Add a zero if necessary = Put coefficients in the box, bring the first one down 7 Put the number you want substituted in on the outside of the box 3 r Multiply the outside numbers, write the answer in the next available spot inside the box 7 21 Add the inside numbers 3 7 21 23 add 3 7 21 23 69 69 207 202 606 597 ANSWER
Synthetic Division Algorithm: 1. Write the dividend in standard form. (Exponents in descending order.) 2. Solve the divisor for x. 3. Use synthetic substitution to determine if k is a zero of the polynomial function. 4. The result will be the coefficients of the quotient. Examples: Summary:
Synthetic Division Solve the denominator for x 3 3 Put this number in front remainder ANSWER Solve the denominator for x 2 2 Put this number in front r.1 or x 3 x 2 x c remainder 03 2 1 ANSWER Summary:
Rational Zeros Algorithm: 1. List all possible rational zeros. The numerator represents all possible factors of the constant term. The denominator represents all possible factors of the leading coefficient. 2. Test all possible zeros using synthetic division. 3. Continue testing zeros until the result is a polynomial that can be factored by grouping, or a trinomial that can be factored into two binomials. 4. If the resulting quadratic cannot be factored, use the quadratic formula to find the remaining zeros. 5. List the real zeros of the function. Example: P4
Finding Zeros Given one zero, find the others ; divide rewrite factor solve ANSWERS Summary:
Real Zeros Find all real zeros of the function. Use your calculator to find one root now use that to verify the other zeros (roots) using synthetic division
Write the Equation of the Polynomial (Leading Coefficient is One) Algorithm: 1. Write the given roots (intercepts) in factored form x = 2 becomes (x 2) 2. Use the distributive Property to multiply 3. Write the function in standard form * Remember all i 's must occur in pairs Examples:
Write the Equation of the Polynomial (Leading Coefficient is NOT One) Algorithm: 1. Write the given roots (intercepts) in factored form x = 2 becomes (x 2) 2. Substitute all roots, x and y into f(x), solve for a. 3. Use the distributive Property to multiply 4. Write the function in standard form * Remember all i 's must occur in pairs Examples:
Graphing Polynomial Functions Algorithm: 1. Find all rational zeros. Write the function in intercept form. 2. Determine the end behavior of the function. 3. Graph all rational zeros. 4. Find all turning points a. Turning points fall half way between two consecutive zeros. This will produce the x value of the turning point. b. Substitute the x value into the original function and solve for the y value. 5. Plot the turning points. 6. Draw a smooth curve through the points. Example:
Use you calculator Graphing Polynomials Find the zeros b61 Find the local maximums and minimums b62 Zero must lie within shaded region Sketch the graph Minimum must lie within shaded region Plot the zeros Plot the local maximums or minimums Connect the points Summary: