This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand.
In these cases, we can take advantage of graphing utilities. We can check whether these are correct by substituting these values for x and verifying that. Each x -intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.
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The test to see if the trinomial is factorable can usually be done mentally. We illustrate by examples. We consider all pairs of factors whose product is 4. Since 4 is positive, only positive integers need to be considered. The possibilities are 4, 1 and 2, 2. We consider all pairs of factors whose product is 3. Since the middle term is positive, consider positive pairs of factors only. The possibilities are 3, 1. We write all possible arrangements of the factors as shown. We select the arrangement in which the sum of products 2 and 3 yields a middle term of 8x.
The integers 4 and -3 have a product of and a sum of 1, so the trinomial is factorable. We now proceed. We consider all pairs of factors whose product is 6. Since 6 is positive, only positive integers need to be considered. Then possibilities are 6, 1 and 2, 3. We consider all pairs of factors whose product is The possibilities are 2, -1 and -2, 1. We write all possible arrange ments of the factors as shown. We select the arrangement in which the sum of products 2 and 3 yields a middle term of x.
With practice, you will be able to mentally check the combinations and will not need to write out all the possibilities. Paying attention to the signs in the trinomial is particularly helpful for mentally eliminating possible combinations. Solution Rewrite each trinomial in descending powers of x and then follow the solutions of Examples 3 and 4. As we said in Section 4. We know that the trinomial is factorable because we found two numbers whose product is 12 and whose sum is 8.
Those numbers are 2 and 6. This is the same result that we obtained before. Some polynomials occur so frequently that it is helpful to recognize these special forms, which in tum enables us to directly write their factored form. Observe that. Often we must solve equations in which the variable occurs within parentheses. We can solve these equations in the usual manner after we have simplified them by applying the distributive law to remove the parentheses.
Parentheses are useful in representing products in which the variable is contained in one or more terms in any factor.
One integer is three more than another. If x represents the smaller integer, represent in terms of x. The larger integer. Five times the smaller integer. Five times the larger integer. Let us say we know the sum of two numbers is If we represent one number by x, then the second number must be 10 - x as suggested by the following table. In general, if we know the sum of two numbers is 5 and x represents one number, the other number must be S - x.
The next example concerns the notion of consecutive integers that was consid- ered in Section 3. The difference of the squares of two consecutive odd integers is The larger integer b. The square of the smaller integer c. The square of the larger integer. Sometimes, the mathematical models equations for word problems involve parentheses. We can use the approach outlined on page to obtain the equation.
Then, we proceed to solve the equation by first writing equivalently the equation without parentheses. So how do I get a nice answer like the ones listed above? Jacob Wheeler Jacob Wheeler 1 1 gold badge 1 1 silver badge 7 7 bronze badges.
Add a comment. Active Oldest Votes. If you clear the denominators in Cymath's answer, you get Wolfram's answer. Solution 2: Complete the square. The Substitute The Substitute 3, 3 3 gold badges 25 25 silver badges 50 50 bronze badges. Sign up or log in Sign up using Google.
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