Question

How do you use synthetic substitution to find g(-5) and g(4) for `g(x)= 9x^4 - 8x^2 + 9x - 5`?

Answer 1

`g(-5)=color(blue)(5375); color(white)(X)g(4)=color(white)(X)color(blue)(2207)`

Explanation:

`g(x) = 9x^4-8x^2+9x-5`

The Remainder Theorem states that when we divide a polynomial `g(x)` by `x-c` the remainder `R` equals `f(c)`.

So, we use synthetic substitution to divide `g(x)` by `x-c`, where `c = -5`.

Step 1. Write only the coefficients of `x` in the dividend inside an upside-down division symbol.

`color(white)(Xll)|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(XX)|`
`color(white)(XX)stackrel("———————————————————)`

Step 2. Put the divisor `(-5)` at the left.

`color(red)(-5)|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(XX)|`
`color(white)(XX)stackrel("———————————————————)`

Step 3. Drop the first coefficient of the dividend below the division symbol.

`-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(XX)|`
`color(white)(XX)stackrel("———————————————————)`
`color(white)(Xll)|color(red)(9)`

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

`-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(Xll)|color(white)(Xl)color(red)(-45)`
`color(white)(XX)stackrel("———————————————————)`
`color(white)(Xll)|9`

Step 5. Add down the column.

`-5|9color(white)(XXl)0color(white)(|l)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(Xll)|color(white)(ll)-45`
`color(white)(XX)stackrel("———————————————————)`
`color(white)(Xll)|9color(white)(ll)color(red)(-45)`

Step 6. Repeat Steps 4 and 5 until you can go no farther

`-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5`
`color(white)(Xll)|color(white)(ll)-45" "225color(white)(l)-1085color(white)(Xl)5380`
`color(white)(XX)stackrel("———————————————————)`
`color(white)(Xll)|9color(white)(l)-45" "217color(white)(l)-1076" "color(red)(5375)`

∴ `g(-5) = 5375`

Check:

`g(x) = 9x^4-8x^2+9x-5`

`g(-5) = 9(-5)^4 -8(-5)^2+9(-5)-5 = 9(625)-8(25)-45-5= 5625-200-50 = 5375`

It works!

Now use the same method to show that

`g(4) = 2207`