Select a factor of the determinant of this matrix.
In order to select a factor of the determinant, we need to find the determinant. We’ll use this first row, follow a positive, negative, positive pattern; we need to take a positive 𝑥 plus four and multiply that times the determinant of this two-by-two matrix. Like this, we’ll subtract that from negative eight times the two-by-two matrix that excludes its column and row. The two-by-two matrix we’ll multiply negative eight by is 𝑥 plus one over four, five over 𝑥 plus eight.
Then we’ll add 𝑥 minus eight times the two-by-two matrix that excludes its row and column. The rest of our steps will be completely algebra, but we’ll have to follow them carefully and make sure that we’re using correct signs. Here we go! Starting on the left, 𝑥 minus four times 𝑥 plus one times 𝑥 plus eight minus five times eight. And before we go on to the next step, let’s go ahead and completely simplify this expression.
We need to distribute 𝑥 plus one over 𝑥 plus eight. 𝑥 times 𝑥 equals 𝑥 squared. 𝑥 times eight equals positive eight 𝑥. One times 𝑥 equals positive one 𝑥. One times eight equals positive eight. Combining the like terms here, 𝑥 squared plus nine 𝑥 plus eight. If we go back up to the top, we’ll notice that we have five times eight there. Five times eight equals 40. So we want to subtract 40. We can’t forget our 𝑥 minus four term that’s hanging out up there.
We have 𝑥 minus four times the determinant of that two-by-two matrix, which was 𝑥 squared plus nine 𝑥 minus 32. We need to multiply this 𝑥 minus four term. 𝑥 times 𝑥 squared equals 𝑥 cubed. 𝑥 times nine 𝑥 equals nine 𝑥 squared. 𝑥 times negative 32, negative 32𝑥. Negative four times 𝑥 squared, negative four 𝑥 squared. Negative four times nine 𝑥, negative 36𝑥. And negative four times negative 32 equals 128. We can simplify again by combining like terms.
There’s no other 𝑥 cubed term, so we bring that down. We have two 𝑥 squared terms. Nine 𝑥 squared minus four 𝑥 squared equals positive five 𝑥 squared. We have two 𝑥 terms. Negative 32 plus negative 36 equals negative 68𝑥. And there’s nothing to combine with 128. So we just bring that down. This is the first of the three times we need to do this process. Onto our negative eight, notice that we’re subtracting negative eight. So we can make that positive eight.
And we’re gonna multiply that positive eight by the determinant 𝑥 plus one times 𝑥 plus eight minus four times five. We start inside our brackets. 𝑥 times 𝑥 equals 𝑥 squared. 𝑥 times eight, eight 𝑥. One times 𝑥, one 𝑥. And one times eight equals eight. Combine like terms. 𝑥 squared plus nine 𝑥 plus eight. Our next multiplication is four times five, which equals 20. We’ll be subtracting 20 from 𝑥 squared plus nine 𝑥 plus eight. Now we have 𝑥 squared plus nine 𝑥 minus 12.
We have that positive eight waiting for us to multiply. Distribute the eight across all three terms. Eight times 𝑥 squared, eight 𝑥 squared. Eight times nine 𝑥, 72𝑥. Eight times 12, negative 96. That’s the second piece of our determinant, now onto the third. We’re taking 𝑥 minus eight and multiplying it by the determinant of the two-by-two matrix 𝑥 plus one times eight minus 𝑥 plus one times four. Working inside the brackets, eight times 𝑥 equals eight 𝑥. Eight times one equals eight. On the other side, four times 𝑥 equals four 𝑥. Four times one equals four.
We’re subtracting four 𝑥 plus four from eight 𝑥 plus eight. We have to make sure that we distribute that negative sign to the four 𝑥 and to the four. Combine like terms. Eight 𝑥 minus four 𝑥 equals positive four 𝑥. Positive eight minus four equals four. We still have this 𝑥 minus eight term we need to multiply. 𝑥 times four 𝑥 equals four 𝑥 squared. 𝑥 times four equals plus four 𝑥. Negative eight times four 𝑥 equals negative 32𝑥. And negative eight times four equals negative 32. And this is the third piece of our determinant.
We now need to take each of these pieces and add them together. 𝑥 cubed plus five 𝑥 squared minus 68𝑥 plus 128 plus eight 𝑥 squared plus 72𝑥 minus 96 plus four 𝑥 squared plus four 𝑥 minus 32𝑥 minus 32.
We’re almost finished, but we need to combine like terms to simplify this expression. There’s only one 𝑥 cubed term, so that can stay. After that, we’ll combine all the 𝑥 squared terms. Five 𝑥 square plus eight 𝑥 squared plus four 𝑥 squared equals 17𝑥 squared. Now we’ll look for all the 𝑥 terms. Negative 68𝑥, 72𝑥, four 𝑥, and negative 32𝑥 equals negative 24𝑥. And last, we’ll take all the constant values: positive 128, negative 96, and negative 32. 128 minus 96 minus 32 equals zero. And that means the determinant here is 𝑥 cubed plus 17𝑥 squared minus
24 [24𝑥]. And each term in the determinant has 𝑥 as a factor. We can remove an 𝑥 and then the determinant would look like this: 𝑥 times 𝑥 squared plus 17𝑥 minus 24. What is the factor of the determinant here? 𝑥 is a factor of this determinant.