7. 3x^3+24x^2-6x-48 10. 27k^3-512 9. 9n^4-100

7. To factor the polynomial $3x^{3}+24x^{2}-6x-48$, we can first factor out the greatest common factor (GCF term. The GCF is 3, so we have:<br /><br />$3(x^{3}+8x^{2}-2x-16)$<br /><br />Next, we can use factoring by grouping. We group the terms in pairs and factor out the GCF from each pair:<br /><br />$3(x^{3}+8x^{2})+(-2x-16)$<br /><br />Factoring out $x^{2}$ from the first group and -2 from the second group, we get:<br /><br />$3(x^{2}(x+8))-2(x+8)$<br /><br />Now we can factor out the common binomial factor $(x+8)$:<br /><br />$3(x^{2}-2)(x+8)$<br /><br />So the factored form of the polynomial $3x^{3}+24x^{2}-6x-48$ is $3(x^{2}-2)(x+8)$.<br /><br />10. To factor the polynomial $27k^{3}-512$, we can recognize that it is a difference of cubes. The general formula for factoring a difference of cubes is:<br /><br />$a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$<br /><br />In this case, we have $a=3k$ and $b=8$, so we can apply the formula:<br /><br />$(3k-8)((3k)^{2}+(3k)(8)+(8)^{2})$<br /><br />Simplifying the terms inside the parentheses, we get:<br /><br />$(3k-8)(9k^{2}+24k+64)$<br /><br />So the factored form of the polynomial $27k^{3}-512$ is $(3k-8)(9k^{2}+24k+64)$.<br /><br />9. To factor the polynomial $9n^{4}-100$, we can recognize that it is a difference of squares. The general formula for factoring a difference of squares is:<br /><br />$a^{2}-b^{2}=(a-b)(a+b)$<br /><br />In this case, we have $a=3n^{2}$ and $b=10$, so we can apply the formula:<br /><br />$(3n^{2}-10)(3n^{2}+10)$<br /><br />So the factored form of the polynomial $9n^{4}-100$ is $(3n^{2}-10)(3n^{2}+10)$.