Domanda del libro di testo
Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places.
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 1, Problema 90
Factor completely, or state that the polynomial is prime.
Guida verificata passo dopo passo1
Identify the quadratic polynomial to factor: \(x^2 - 11x + 28\).
Recall that to factor a quadratic of the form $x^2 + bx + c$, we look for two numbers that multiply to \(c\) and add to \(b\).
Find two numbers that multiply to \(28\) and add to \(-11\). Consider the factors of \(28\): \(1 \times 28\), \(2 \times 14\), \(4 \times 7\).
Since the sum is negative and the product is positive, both numbers must be negative. Check if \(-4\) and \(-7\) work: \(-4 \times -7 = 28\) and \(-4 + (-7) = -11\).
Write the factored form using these numbers: \((x - 4)(x - 7)\).
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Factoring Quadratic Polynomials
Factoring quadratic polynomials involves expressing a quadratic expression as a product of two binomials. For a quadratic in the form ax^2 + bx + c, we look for two numbers that multiply to ac and add to b. This process simplifies solving equations and analyzing polynomial properties.
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Introduction to Factoring Polynomials
Prime Polynomials
A prime polynomial is one that cannot be factored into polynomials of lower degree with coefficients in the given number system. Recognizing when a quadratic is prime helps determine if factoring is possible or if the polynomial must be left as is.
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07:30Introduction to Factoring Polynomials
The Relationship Between Coefficients and Roots
The coefficients of a quadratic polynomial relate directly to its roots through the sum and product: the sum of the roots equals -b/a, and the product equals c/a. Understanding this helps identify factor pairs quickly when factoring quadratics.
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02:20Imaginary Roots with the Square Root Property
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