اختبار اختيار من متعدد تفاعلي: Solving Polynomial Equations by Graphing and Factoring - Reveal
Polynomial equations are algebraic expressions consisting of variables and coefficients, where the highest power of the variable determines the degree of the polynomial. Solving these equations often involves finding the roots or zeros, which can be achieved through various methods such as factoring, graphing, or using the quadratic formula. Factoring is a fundamental technique where the polynomial is broken down into simpler 'factors' that, when multiplied, yield the original expression. Common factoring methods include identifying the Greatest Common Factor (GCF), grouping terms, and recognizing special products like the difference of squares or perfect square trinomials. Graphing provides a visual representation where the solutions correspond to the x-intercepts of the function.
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To factor p2 - 14p, identify the greatest common factor (GCF). Both terms contain at least one p. Factoring out p from p2 leaves p, and factoring it out from -14p leaves -14, resulting in p(p - 14).
Question 2
Points: 1
Factor by Grouping: x3 - 5x2 + 5x - 25
Explanation
Group the first two terms and the last two terms: (x3 - 5x2) + (5x - 25). Factor out the GCF from each group: x2(x - 5) + 5(x - 5). Now, factor out the common binomial (x - 5) to get (x2 + 5)(x - 5).
Question 3
Points: 1
Factor Completely: 3n2 - 15n + 18
Explanation
First, factor out the GCF of 3: 3(n2 - 5n + 6). Then, factor the quadratic trinomial n2 - 5n + 6 by finding two numbers that multiply to 6 and add to -5, which are -3 and -2. This gives 3(n - 3)(n - 2).
Question 4
Points: 1
What is the first step in solving this quadratic equation? x2 + 5x = 6
Explanation
To solve a quadratic equation using factoring or the quadratic formula, the equation must be in standard form (ax2 + bx + c = 0). Therefore, the first step is to subtract 6 from both sides.
Question 5
Points: 1
What are the 2 characteristics we look for when trying to solve a quadratic equation?
Explanation
The standard procedure for solving quadratic equations involves setting the entire expression to zero and, for easier factoring, ensuring the leading coefficient (the coefficient of x2) is positive.
Question 6
Points: 1
What should you do first in solving this equation? x2 + 6x - 13 = 3
Explanation
Similar to standard form requirements, before applying factoring or solving techniques, the equation must be set to zero by moving all terms to one side.
Question 7
Points: 1
Convert x2 - 9x + 20 = 0 to factored form and solve.
Explanation
To factor x2 - 9x + 20, look for two numbers that multiply to 20 and add to -9. Those numbers are -4 and -5. Thus, the factored form is (x - 4)(x - 5) = 0, giving solutions x = 4 and x = 5.
Question 8
Points: 1
Factor: 49x2 - 100
Explanation
This is a difference of squares problem following the pattern a2 - b2 = (a - b)(a + b). Since 49x2 = (7x)2 and 100 = 102, it factors as (7x - 10)(7x + 10).
Question 9
Points: 1
Factor x2 + 6x + 9 = 0 and solve for x.
Explanation
This is a perfect square trinomial following the pattern a2 + 2ab + b2 = (a + b)2. Here, x2 + 2(3)(x) + 32 factors into (x + 3)2 = 0, which leads to the single root x = -3.
Question 10
Points: 1
Factor 6x2 - 27x
Explanation
Identify the greatest common factor (GCF) for the coefficients 6 and 27, which is 3, and for the variable terms x2 and x, which is x. Factoring out 3x gives 3x(2x - 9).
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