Domanda del libro di testo
In Exercises 23–34, factor out the negative of the greatest common factor.
−x² − 7x + 5
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0. Review of Algebra
Factoring Polynomials
1:32 minutiProblema 35
Domanda del libro di testo
In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 4a² + 12x + 36
Guida verificata passo dopo passo1
Step 1: Begin by rearranging the terms of the polynomial to group them in a way that might reveal a common factor or pattern. Consider the expression: \(x^2 + 12x - 4a^2 + 36\).
Step 2: Notice that the first three terms \(x^2 + 12x + 36\) form a perfect square trinomial. Rewrite it as \((x + 6)^2\).
Step 3: Substitute the perfect square trinomial back into the expression, resulting in \((x + 6)^2 - 4a^2\).
Step 4: Recognize that the expression is now in the form of a difference of squares, \((A^2 - B^2)\), where \(A = (x + 6)\) and \(B = 2a\).
Step 5: Apply the difference of squares formula \(A^2 - B^2 = (A - B)(A + B)\) to factor the expression as \((x + 6 - 2a)(x + 6 + 2a)\).
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Factoring Polynomials
Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor, using special products like the difference of squares, and applying the quadratic formula when necessary.
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Quadratic Expressions
A quadratic expression is a polynomial of degree two, typically in the form ax² + bx + c. Understanding the structure of quadratic expressions is crucial for factoring, as it allows one to identify potential roots and apply methods such as completing the square or using the quadratic formula. Recognizing patterns in quadratics can also facilitate easier factoring.
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Prime Polynomials
A prime polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials with real coefficients. Identifying whether a polynomial is prime is important in algebra, as it determines the methods available for solving equations. A polynomial may be prime if it does not have rational roots or if it cannot be expressed in simpler polynomial forms.
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