Phân tích đa thức sau thành nhân tử: ab (a^2 - b^2) + bc (b^2 - c^2) + ca (c^2 - a^2)
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\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(ab\left(a-b\right)+bc\left(b-c\right)+ca\left(c-a\right)\)
\(=ab\left(a-b\right)+bc\left(b-c\right)-ca\left(a-c\right)\)
\(=ab\left(a-b\right)+bc\left(b-c\right)-ca\left(a-b+b-c\right)\)
\(=ab\left(a-b\right)+bc\left(b-c\right)-ca\left(a-b\right)-ca\left(b-c\right)\)
\(=\left(a-b\right)\left(ab-ca\right)+\left(b-c\right)\left(bc-ca\right)\)
\(=\left(a-b\right)a\left(b-c\right)+\left(b-c\right)c\left(b-a\right)\)
\(=\left(a-b\right)a\left(b-c\right)-\left(b-c\right)c\left(a-b\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\)
mình làm vội, có chỗ nào sai bạn thông cảm nha
![](https://rs.olm.vn/images/avt/0.png?1311)
ai có thể giảng cho mình dạng toán tìm số tự nhiên thỏa mãn đièu kiện chia hết ko
hãy nêu ra cách giải cụ thể cho câu sau 3a-11 chia hết cho a+2 tìm a
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=a^2b-ab^2+b^2c-bc^2+ac^2-a^2c\)
\(=a^2\left(b-c\right)+bc\left(b-c\right)-a\left(b-c\right)\left(b+c\right)\)
\(=\left(b-c\right)\left(a^2-bc-ab-ac\right)\)
\(=\left(b-c\right)\left[a\left(a-b\right)-c\left(a-b\right)\right]\)
\(ab\left(a^2-b^2\right)+bc\left(b^2-c^2\right)+ca\left(c^2-a^2\right)\)
\(=a^3b-ab^3+b^3c-bc^3+c^3a-ca^3\)
\(=a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(=a^3\left(b-a+a-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(=\left(a-b\right)\left(c^3-a^3\right)+\left(c-a\right)\left(b^3-a^3\right)\)
\(=\left(a-b\right)\left(c-a\right)\left(c^2+ca+a^2\right)-\left(c-a\right)\left(a-b\right)\left(a^2+ab+b^2\right)\)
\(=\left(a-b\right)\left(c-a\right)\left(c^2+ca+a^2-a^2-ab-b^2\right)\)
\(=\left(a-b\right)\left(c-a\right)\left(c^2+ca-ab-b^2\right)\)
\(=\left(a-b\right)\left(c-a\right)\left[-a\left(b-c\right)-\left(b-c\right)\left(b+c\right)\right]\)
\(=-\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\)