High School Cluster Quiz Polynomial



Polynomial FunctionsWrite an expression that is equivalent to (3x2-6)+(2x2-4x+7)+(6x-2), combining all like terms.Which expression is equivalent to mx2+a+(nx2+b)?A.mnx2+abB.m+nx2+a+bC.m+nx2+(a+b)D.m+ax2+(n+b)x2Write an expression equivalent to 3y-3x2(-2xy4) in the form Axmyn.Multiply and combine like terms to determine the product of these polynomials.(-2x-3)(2x2-x+1)(x-2)Which expression is equivalent to (4+x)(4x+t)?A.16x+xtB.8x+4x2+tC.8x+5x2+4t+xtD.16x+4x2+4t+xtWhich expression is equivalent to ax(bx+c)?A.abx2+acx2B.abx2+acxC.abx2+cD.abx+cFind all the zeros of the following polynomial function and enter them into the boxes.fx= x4-2x3-28x2+8x+9Kiera claims that the sum of two linear polynomials with rational coefficients is always a linear polynomial with rational coefficients.Use the numbers 1 (first) through 6 (last) to place the statements into a logical sequence to outline an argument that supports Kiera’s claim.StatementOrder (1 is first, 6 is last)px+qx=ax+cx+(b+d)px+qx=ax+b+(cx+d)Given px=ax+b and qx=cx+d where a, b, c, and d are rational numbers.a+c and (b+d) are rational numbersSo px+qx is a linear polynomial with rational coefficients.px+qx=a+cx+(b+d)P(x) is a 4th degree polynomial. The graph of y=P(x) has exactly three distinct x-intercepts.Part AWhich polynomial could be P(x)?A.x3(x-3)B.x2x-2(x-1)C.(x-3)(x-2)(x-1)D.x(x-3)(x-2)(x-1)Part BFor one of the polynomials above, explain why it could not be P(x).Proposition 1 The sum of two linear polynomials with integer coefficients is always a linear polynomial with integer coefficients.The outline of a proof of Proposition 1 is shown. Add one or more justifications for steps 3, 4, and 5 of the proof from the list of justifications below.The Closure of the Integers Under Addition The sum of two integers is always an integerThe Commutative Property If A and B are real numbers, then A + B = B + AThe Associative Property If A, B, and C are real numbers, then (A + B) + C = A + (B + C)The Distributive Property If A, B, and C are real numbers, then A(B + C) = AB + ACThe Any-Order Property of Addition The sum of two or more real numbers can be performed in any order or any grouping.StatementJustificationGiven px=ax+b and qx=cx+d where a, b, c, and d are integers. Hypothesispx+qx=ax+b+(cx+d)By Definitionpx+qx=ax+cx+(b+d)px+qx=a+cx+(b+d)a+c and (b+d) are integersSo px+qx is a linear polynomial with integer coefficients.ConclusionTeacher MaterialA-SSE.AInterpret the structure of expressions.A-APR.APerform arithmetic operations on polynomials.F-IF.AUnderstand the concept of a function and use function notation.F-IF.CAnalyze functions using different representations.F-BF.ABuild a function that models a relationship between two quantities.QuestionClaimKey/Suggested Rubric111 point: 5x2+2x-1, or equivalent expression with 3 terms.2111 point: Selects C.3111 point: Writes -6x-1y.4111 point: Writes, -4x4+4x3+9x2-5x+6, or equivalent expression with 5 terms.5111 point: Selects D.6111 point: Selects B.721 point: x = –4, x = 6, x = –2, and x = 2 (in any order).NOTE: Allow students to just write the values of x (–4, 6, –2, and 2) in the boxes.831 point:StatementOrder (1 is first, 6 is last)px+qx=ax+cx+(b+d)3px+qx=ax+b+(cx+d)2Given px=ax+b and qx=cx+d where a, b, c, and d are rational numbers.1a+c and (b+d) are rational numbers5So px+qx is a linear polynomial with rational coefficients.6px+qx=a+cx+(b+d)49332 points: Selects B AND Explains why A, C, or D could not be P(x). Example 1: A could not be P(x) because its graph only has two distinct x-intercepts at 0 and 3. Example 2: C could not be P(x) because it’s only a 3rd degree polynomial. Example 3: D could not be P(x) because its graph has 4 distinct x-intercepts at 0, 3, 2, and 1.1 point: Selects B OR Explains why A, C, or D could not be P(x).10331 point:StatementJustificationGiven px=ax+b and qx=cx+d where a, b, c, and d are integers.Hypothesispx+qx=ax+b+(cx+d)By Definitionpx+qx=ax+cx+(b+d)The Any-Order Property of Addition ORThe Commutative Propertypx+qx=a+cx+(b+d)The Distributive Propertya+c and (b+d) are integersThe Closure of the Integers Under AdditionSo px+qx is a linear polynomial with integer coefficients.Conclusion ................
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