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1st degree polynomials are called linear and can be put into the form ax+b. Linear polynomials have at least one real root.
An 2nd degree polynomials are called quadratic and can be put into the form ax²+bx+c. If a quadratic polynomial can be facorted, the factors are linear. A quadratic polynomial has at most two real roots.
Using a process called completing the square , the quadratic formula for the roots of a quadratic polynomial was derived. The student is referred to page 114 of the text for another look at this important derivation. You will find that the technique of completing the square is required throughout this course. Learn it now! In this lecture we used the technique to find that x²-3x+7=0 has no real solutions.
Another important idea which completing the square brings to light is the discriminant D of a quadratic expression. It is given by D=b²-4ac, where b is the coefficient of the linear term, a is the coefficient of the square term, and c is the constant term. You can put each quadratic expression into one of the six categories: A. two real roots, opens upward; B. one (repeated) root, opens upward; C. No real roots, opens upward; D. two real roots, opens downward; E. one (repeated) root, opens downward; and F. No real roots, opens downward. You are expected to be able to carry out this classification.
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