What degree is the product of a polynomial of degree

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81. What degree is the productof a polynomial of degree nand apolynomial of degree m? 82.What degree is the sumof a polynomial of degree nand apolynomial of degree m? m > n 83. (7 x 4 y 2 ) 2 (7 x 4 y 2 ) 2 84. (3 x 5 y 2 ) 2 (3 x 5 y 2 ) 2 85. ( x a )( x 2 ax a 2 ) 86. ( x a )( x 2 ax a 2 ) 87. Use a graphing utility to plot the graphs of the three expressions , and Which two graphs agree with each other? 2 x 2 - 5 x - 12. (2 x + 3)( x - 4), 2 x 2 + 5 x - 12 88.Use a graphing utility to plot the graphs of the three expressions Whichtwo graphs agree with each other? ( x + 5) 2 , x 2 + 25, and x 2 + 10 x + 25. SECTION 0.4 FACTORING POLYNOMIALS 75. Subtract and simplify Solution: Eliminate the parentheses. 2 x 2 5 3 x x 2 1 Collect like terms. x 2 3 x 4 This is incorrect. What mistake was made? (2 x 2 - 5) - (3 x - x 2 + 1). 76. Simplify (2 x ) 2 . Solution: Write the square of the binomial as the sum of the squares. (2 x ) 2 2 2 x 2 Simplify. x 2 4 This is incorrect. What mistake was made? In Exercises 75 and 76, explain the mistake that is made. C ATC H T H E M I S TA K E C O N C E P T U A L C H A L L E N G E T E C H N O L O GY
To factor the resulting polynomial, you reverse the process to undo the multiplication: The polynomials ( x 3) and ( x 1) are called factors of the polynomial x 2 4 x 3. The process of writing a polynomial as a product is called factoring . In Chapter 1 we will solve quadratic equations by factoring. In this section we will restrict our discussion to factoring polynomials with integer coefficients, which is called factoring over the integers . If a polynomial cannot be factored using integer coefficients, then it is prime or irreducible over the integers. When a polynomial is written as a product of prime polynomials, then the polynomial is said to be factored completely . Greatest Common Factor The simplest type of factoring of polynomials occurs when there is a factor common to every term of the polynomial. This common factor is a monomial that can be “factored out” by applying the distributive property in reverse: For example, 4 x 2 6 x can be written as 2 x ( x ) 2 x (3). Notice that 2 x is a common factor to both terms, so the distributive property tells us we can factor this polynomial to yield 2 x ( x 3). Although 2 is a common factor and x is a common factor, the monomial 2 x is called the greatest common factor .

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