Video: Using Synthetic Substitution to Evaluate Polynomial Functions

Use synthetic substitution to find the value of 𝑓(20) given 𝑓(𝑥) = 0.06𝑥⁴ − 0.14𝑥³ − 3.1𝑥 + 5.4.

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Video Transcript

Use synthetic substitution to find the value of 𝑓 of 20 given 𝑓 of 𝑥 equals 0.06𝑥 to the fourth minus 0.14𝑥 cubed minus 3.1𝑥 plus 5.4.

To set up synthetic substitution, we start with the value of 𝑥 that we’re trying to solve for, in this case 20. And then, we take the coefficients from each term: 0.06 from 𝑥 to the fourth, negative 0.14 from 𝑥 cubed. Now, this is important. Notice that we’re adding the coefficients from each 𝑥 term with descending exponent values. We have to do this for every 𝑥 term.

The function we’re given does not have a coefficient with 𝑥 squared. So we add a zero there as a placeholder to represent the 𝑥 squared term. From there, we have negative 3.1𝑥 and 𝑥 to the zero power would be the constant 5.4.

And now, we’re ready to solve. The first step is to bring down our 0.06. Then, we’re going to multiply 20 by 0.06. 20 times 0.06 is 1.2. Now, we’re going to add negative 0.14 and 1.2 which equals 1.06. Now, we multiply 20 times 1.06 which equals 21.2. 21.2 plus zero equals 21.2. Multiply 21.2 by 20 and we get 424. 424 plus negative 3.1 equals 420.9. Multiply this value, 420.9 by 20 and we get 8418. Add 5.4 to that value and you get 8423.4.

The value of the function given at 20 equals 8423.4. Pretty cool, right?

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