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Additional info for CSET Mathematics 110, 111, 112 (XAM CSET)
Example text
2 (12 ) + 1 + 7 = 25 + 7 = 5 + 7 = 12 2 ( 4 ) + 1 + 7 = 9 + 7 = 3 + 7 = 10 ≠ 4 Only x = 12 satisfies the original expression. This is therefore the only solution to the equation. Plotting a radical function follows a process similar to that of plotting virtually any other function. A set of representative points is needed, and prior knowledge of the domain of the function is helpful (for instance, if only real numbers are considered, the function in the previous example has only non-negative numbers in its domain).
One method of solving it is by factoring the quadratic expression and applying the condition that at least one of the factors must equal zero in order for the whole expression to be zero. The proof of the quadratic formula uses the following approach. ax 2 + bx + c = 0 b c x2 + x + = 0 a a b c x2 + x = − a a Complete the square of the left-hand side of the equation. 2 b c b b x + x + = − + a a 2a 2a 2 2 Factor and take the square root while simultaneously simplifying the right-hand side of the equation.
Then: To prove the Rational Root Theorem, first assume that x = n p p p P= an + an −1 q q q n −1 2 p p + + a2 + a1 = + a0 0 q q Multiply the entire expression by q n . an p n + an −1p n −1q + + a2 p 2q n −2 + a1pq n −1 + a0q n = 0 MATHEMATICS 110, 111, 112 15 TEACHER CERTIFICATION STUDY GUIDE Since each coefficient ai is an integer, as are p and q, each term in the above expression must also be an integer. Consequently, any partial sum of the terms is an integer as well.