Dividing Polynomials VOCABULARY


 Laurence Cunningham
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1  Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS ()(G) VOCABULARY Synthetic division Synthetic division is a process for dividing a polynomial by a linear expression x  a. You list the standardform coefficients (including zeros) of the polynomial, omitting all variables and exponents. You use a for the divisor and add instead of subtract throughout the process. Apply use knowledge or information for a specific purpose, such as solving a problem ESSENTIAL UNDERSTANDING You can divide polynomials using steps that are similar to the longdivision steps that you use to divide whole numbers. Key Concept The Division Algorithm for Polynomials You can divide polynomial P (x) by polynomial D (x) to get polynomial quotient Q (x) and polynomial remainder R (x). The result is P (x) = D (x)q (x) + R (x). Q (x) D (x))p (x) # ## R (x) If R (x) =, then P (x) = D (x)q (x) and D (x) and Q (x) are factors of P (x). To use long division, P (x) and D (x) should be in standard form with zero coefficients where appropriate. The process stops when the degree of the remainder, R (x), is less than the degree of the divisor, D (x). Theorem The Remainder Theorem If you divide a polynomial P (x) of degree n Ú by x  a, then the remainder is P (a). PearsonTEXAS.com
2 Problem Using Polynomial Long Division Use polynomial long division to divide x + x x by x. What is the quotient and remainder? How can you check your result? Show that (divisor)(quotient) + remainder = dividend. x x  )x + x  x  x  x x  Divide: x x = x. Multiply: x(x  ) = x  x. Subtract to get x. Bring down . Repeat the process of dividing, multiplying, and subtracting. x + x  )x + x  x  x  x x  x   The quotient is x + with remainder . Check (x  )(x + )  = (x + x  x  )  = x + x  x  Divide: x x = Multiply: (x  ) = x . Subtract to get . You can say: x, R. Multiply (x  )(x + ). Simplify. Problem TEKS Process Standard ()(G) Using Polynomial Long Division to Check Factors A Use polynomial long division to divide P(x) = x x + x + by x +. Is x + a factor of P(x)? x  x + x + x + )x  x + x + x + x + x + x x + x + x x + x  x x + x + x + x + x  Include x terms. The degree of the remainder is less than the degree of the divisor. Stop! The remainder is not zero. x + does not divide x  x + x + evenly and is not a factor of P(x). continued on next page Lesson  Dividing Polynomials
3 Problem continued Can you use the Factor Theorem to help answer this question? Yes; recall that if P(a) =, then x  a is a factor of P(x). B Is x a factor of P (x) = x? If it is, write P (x) as a product of two factors. Step Use the Factor Theorem to determine if x  is a factor of x . P () =  =  = Since P () =, x  is a factor of P (x). Step Use polynomial long division to find the other factor. x + x + x + x  )x + x + x + x  x  x x + x x  x x + x x  x x  x  P (x) = (x  )(x + x + x + ) Problem Using Synthetic Division To divide by x + what number do you use for the synthetic divisor? x + = x  (), so use . Use synthetic division to divide x x + x x by x +. What is the quotient and remainder? Step Reverse the sign of +. Write the coefficients of the polynomial.  ; Step Multiply the coefficient by the divisor. Add to the next coefficient.  ; Step Bring down the first coefficient.  ; Step Continue multiplying and adding through the last coefficient.  ; The quotient is x  x + x , R. PearsonTEXAS.com
4 Problem TEKS Process Standard ()(A) How can you use the picture to help solve the problem? The picture gives the width of the box. Remember for a rectangular prism, V = / * w * h. Using Synthetic Division to Solve a Problem Crafts The polynomial x + x x expresses the volume, in cubic inches, of the shadow box shown. A What are the dimensions of the box? (Hint: The length is greater than the height (or depth).)  ; x + x  = (x  )(x + ) So, x + x  x  = (x + )(x + x  ) = (x + )(x  )(x + ) The length, width, and height (or depth) of the box are (x + ) in., (x + ) in., and (x  ) in., respectively. B If the width of the box is in., what are the other two dimensions? The width of the box is x +. So if x + =, then x =. Substitute for x to find the length and height (or depth). Length: x + = + = in. Height: x  =  = in. x + Is there a way to find P() without substituting? Use synthetic division. P() is the remainder. Problem Evaluating a Polynomial Given that P (x) = x x x +, what is P ()? By the Remainder Theorem, P () is the remainder when you divide P (x) by x . P () =. ; Lesson  Dividing Polynomials
5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Divide using long division. Check your answers. For additional support when completing your homework, go to PearsonTEXAS.com.. x + x , (x + ). x + x  x +, (x  ). x  x  x , (x  ). x + x  x , (x  ) Divide.. x + x + x +, (x + ). x + x + x +, (x + ). x + x  x , (x  ). x  x + x + x , (x  ) Determine whether each binomial is a factor of x + x + x.. x +. x +. x +. x  Divide using synthetic division.. x + x  x , (x  ). x  x + x , (x  ). x  x  x +, (x + ). x  x  x , (x  ). x +, (x  ). x + x + x , (x + ). x +, (x + ). x  x , (x  ) Use synthetic division and the given factor to completely factor each polynomial function.. y = x + x  x  ; (x + ). y = x  x  x + ; (x + ). Apply Mathematics ()(A) The volume, in cubic inches, of the decorative box shown can be expressed as the product of the lengths of its sides as V (x) = x + x  x. What linear expressions with integer coefficients represent the length and height of the box? x Use synthetic division and the Remainder Theorem to find P (a).. P (x) = x + x  x  ; a =. P (x) = x + x + x; a =. P (x) = x  x + x  ; a =. P (x) = x + x + x; a =. P (x) = x  x + x + ; a =. P (x) = x  x + x + ; a =. P (x) = x + x  x  ; a =. P (x) = x + x + x  ; a = PearsonTEXAS.com
6 . Select Techniques to Solve Problems ()(C) Your friend multiplies x + by a quadratic polynomial and gets the result x  x  x +. The teacher says that everything is correct except for the constant term. Find the quadratic polynomial that your friend used. What is the correct result of multiplication?. Display Mathematical Ideas ()(G) A student used synthetic division to divide x  x  x by x +. Describe and correct the error shown.. Connect Mathematical Ideas ()(F) When a polynomial is divided by (x  ), the quotient is x + x + with remainder. Find the polynomial.. Apply Mathematics ()(A) The expression (x + x + x + ) represents the volume of a square pyramid. The expression x + represents the height of the pyramid. What expression represents the side length of the base? (Hint: The formula for the volume of a pyramid is V = Bh.). Analyze Mathematical Relationships ()(F) Divide. Look for patterns in your answers. a. x , (x  ) b. x , (x  ) c. x , (x  ) d. Using the patterns, factor x .. Select Tools to Solve Problems ()(C) The remainder from the division of the polynomial x + ax + ax + by x + is. Find a.. Use synthetic division to find (x + ), (x  i).. Display Mathematical Ideas ()(G) Suppose, , and are zeros of a cubic polynomial function f (x). What is the sign of f () # f ()? (Hint: Sketch the graph; consider all possibilities.) TEXAS Test Practice. What is the remainder when x  x + is divided by x +? A. B. C. D.. What is the least degree of a polynomial that has a zero of multiplicity at, a zero of multiplicity at, and a zero of multiplicity at? F. G. H. J.. The equation y =.x relates your weight on the Moon y to your weight on Earth x in pounds. If Al weighs lb on Earth, what would he weigh on the Moon? A.. lb B.. lb C. lb D.. lb. The formula for the area of a circle is A = pr. Solve the equation for r. If the area of a circle is. cm, what is the radius? Use. for p. Lesson  Dividing Polynomials
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