Interactive multiple-choice quiz: Solving Polynomial Equations by Graphing and Factoring - Reveal
This worksheet focuses on various methods for solving and factoring polynomial equations. Topics include factoring by grouping, using the greatest common factor (GCF), factoring differences of squares, and finding the zeros of functions from equations or graphs. It also covers identifying the number of solutions based on the degree of the polynomial and dealing with complex solutions.
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اختبار شهادة تدريبي مؤقت للصف والمادة والفصل نفسه.
اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
To solve 4x2 + 11x - 3 = 0 by factoring, find two numbers that multiply to -12 and add to 11, which are 12 and -1. This gives (4x - 1)(x + 3) = 0, resulting in \(x = \frac{1}{4}\) and x = -3.
Solve by factoring: 3x4 - 6x2 + 3 = 0 (Hint: take out the GCF)
Explanation
Factoring out the GCF: 3(x4 - 2x2 + 1) = 0. This is a perfect square trinomial: 3(x2 - 1)2 = 0. Factoring further: 3((x - 1)(x + 1))2 = 0, which gives roots x = 1 and x = -1, each with multiplicity 2.
How many solutions does the following expression have? 7x5 - 3x4 + 5x - 8
Explanation
The number of solutions is determined by the highest power of the variable (the degree of the polynomial). Since the degree is 5, there are 5 solutions.
Let u = x2, then \(u^2 + u - 90 = 0 \implies (u + 10)(u - 9) = 0\). Thus \(x^2 = 9 \implies x = \pm 3\), and \(x^2 = -10 \implies x = \pm i\sqrt{10}\). Note: The provided answer key chooses option (a), though the square root of -10 is imaginary.
The zeros of the function are the x-intercepts of the graph. Looking at the provided graph, the curve crosses the x-axis at x = -3, x = -1, x = 2, and x = 5.
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