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>n83.(7x4y2)2(7x4y2)284.(3x5y2)2(3x5y2)285.(xa)(x2axa2)86.(xa)(x2axa2)87.Use a graphing utility to plot the graphs of the three expressions , andWhich two graphs agree with each other?2x2-5x-12.(2x+3)(x-4), 2x2+5x-1288.Use a graphing utility to plot the graphs of the three expressions Whichtwo graphs agree with each other?(x+5)2, x2+25, and x2+10x+25.SECTION0.4FACTORING POLYNOMIALS75.Subtract and simplify Solution:Eliminate the parentheses. 2x25 3xx21Collect like terms.x23x4This is incorrect. What mistake was made?(2x2-5)-(3x-x2+1).76.Simplify (2 x)2.Solution:Write the square of the binomial as the sum of the squares.(2 x)222x2Simplify.x24This 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 factorthe resulting polynomial, you reverse the process to undo the multiplication:The polynomials (x3) and (x1) are called factorsof the polynomial x24x3.The process of writing a polynomial as a product is called factoring. In Chapter 1 we willsolve quadratic equations by factoring.In this section we will restrict our discussion to factoring polynomials with integercoefficients, which is called factoring over the integers. If a polynomial cannot be factored using integer coefficients, then it is primeor irreducible over the integers. When apolynomial is written as a product of prime polynomials, then the polynomial is said to befactored completely.Greatest Common FactorThe simplest type of factoring of polynomials occurs when there is a factor common toevery term of the polynomial. This common factoris a monomial that can be “factoredout” by applying the distributive property in reverse:For example, 4x26x can be written as 2x(x) 2x(3). Notice that 2x is a common factorto both terms, so the distributive property tells us we can factor this polynomial to yield2x(x3). Although 2 is a common factor and xis a common factor, the monomial 2x iscalled the greatest common factor.